Econometrica, Vol. 67, No. 2 March, 1999 , 251 Ž. 333

Ž .Econometrica, Vol. 67, No. 2 March, 1999 , 251�333

HIGH WAGE WORKERS AND HIGH WAGE FIRMS

BY JOHN M. ABOWD, FRANCIS KRAMARZ, AND DAVID N. MARGOLIS1

We study a longitudinal sample of over one million French workers from more thanfive hundred thousand employing firms. We decompose real total annual compensationper worker into components related to observable employee characteristics, personalheterogeneity, firm heterogeneity, and residual variation. Except for the residual, allcomponents may be correlated in an arbitrary fashion. At the level of the individual, wefind that person effects, especially those not related to observables like education, are avery important source of wage variation in France. Firm effects, while important, are notas important as person effects. At the level of firms, we find that enterprises that hirehigh-wage workers are more productive but not more profitable. They are also morecapital and high-skilled employee intensive. Enterprises that pay higher wages, controllingfor person effects, are more productive and more profitable. They are also more capitalintensive but are not more high-skilled labor intensive. We find that person effects explainabout 90% of inter-industry wage differentials and about 75% of the firm-size wage effectwhile firm effects explain relatively little of either differential.

KEYWORDS: Wage determination, person effects, firm effects, inter-industry wagedifferentials, heterogeneity.

1. INTRODUCTION

FOR DECADES LABOR ECONOMISTS have lamented the lack of microeconomic dataŽrelating characteristics of firms to characteristics of their workers see, for

Ž . Ž ..example, Rosen 1986 and Willis 1986 because such data would permitresearchers to begin to disentangle the effects of firm-level decisions from theeffects of choices made by workers. Why do high-paying firms provide more thanthe apparent going wage? Perhaps such a strategy delivers a gain in productivityor profitability that exceeds the incremental wage cost, as predicted by efficiencywage and agency models.2 Perhaps high-paying firms select workers with higherexternal wage rates or better firm-specific matches, thus sorting the workers into

1 The authors gratefully acknowledge the financial and computing support of INSEE. TheŽ .National Science Foundation supported Abowd and Margolis SBR 91-11186, 93-21053 and 96-18111 .

We are grateful for the comments of a co-editor, Ronald Ehrenberg, Hank Farber, Robert Gibbons,Guy Laroque, Stefan Lollivier, Bentley MacLeod, Olivia Mitchell, Ariel Pakes, Alain Trognon, and´Martin Wells as well as for comments received during seminars far too numerous to mention here.The data used in this paper are confidential but the authors’ access is not exclusive. Otherresearchers interested in using these data should contact the Centre de Recherche en Economie etStatistique, INSEE, 15 bd Gabriel Peri, 92245 Malakoff Cedex, France.´

2 Ž . Ž . Ž .See Lazear 1979 , Shapiro and Stiglitz 1984 , Hart and Holmstrom 1987 , and Sappington¨Ž .1991 for concise statements of the theories generating these predictions. Tests of these models

Ž . Ž . Ž .have been performed by Abowd 1990 , Abowd and Kramarz 1993 , Cahuc and Dormont 1997 ,Ž . Ž . Ž . Ž .Gibbons and Murphy 1990, 1992 , Hutchens 1987 , Kahn and Sherer 1990 , and Leonard 1990 .

251

J. M. ABOWD, F. KRAMARZ, AND D. N. MARGOLIS252

firms that have differential observed compensation programs.3 Although broadlyrepresentative linked surveys of firms and workers are not available in the U.S.,there have now been numerous studies that attempt to relate firm performanceto the design of the compensation system.4 Furthermore, many have analyzedthe inter-industry wage differentials among individuals as if they were themanifestation of differences in firm level compensation policies.5 In this paperwe present the first extensive statistical analysis of simultaneous individual- andfirm-level heterogeneity in compensation determination. We examine the varia-tion in personal wage rates holding firm effects constant and variation in firm

Žwage rates holding person effects constant. Due to the matched person and.firm longitudinal nature of our data, we are able to control for both measured

and unmeasured heterogeneity in the workers and their employing firms.A high-wage worker is a person with total compensation higher than expected

on the basis of observable characteristics like labor force experience, education,region, or sex. A high-wage firm is an employer with compensation higher thanexpected given these same observable characteristics. Until now all empiricalanalyses of personal and firm heterogeneity in compensation outcomes haverelied upon data that were inadequate to identify separately the individual effectnecessary to classify a worker as high-wage and the firm effect required toclassify a firm as high-wage.

Using a unique longitudinal data set of firms and workers that is representa-tive of private sector French employment, we are able to estimate both personand firm components of compensation determination, allowing for observableand unobservable factors in both dimensions and unrestricted correlation amongthe effects. Computational complexity prevents full least squares estimation ofthe models with unobserved heterogeneity in both the person and firm dimen-sions. After discussing these issues, we examine in detail several related statisti-cal solutions, one of which is a consistent estimator of some of the parameters,and two others that are conditional methods. We also consider other simpler,more classical, techniques in order to assess the importance of person and firmheterogeneity. Although none of these techniques can be used to compute thefull least squares solution to the statistical problem, which, for the moment,remains computationally infeasible, all of our methods approximate the full leastsquares solution and allow the components of person and firm heterogeneity tobe intercorrelated. Our consistent method permits estimation of all time-varyingcoefficients, including those that are heterogeneous. One of our conditionalmethods, called ‘‘order independent,’’ has the advantage that the estimated

3 Ž . Ž . Ž .This view is espoused by Bulow and Summers 1976 , Cain 1976 , Jovanovic 1979 , and RoyŽ . Ž .1951 . Weiss and Landau 1984 present a different theoretical version of this model. Some tests

Ž . Ž . Ž .include Dickens and Lang 1985 , Flinn 1986 , Gibbons and Katz 1991 , and Heckman andŽ .Sedlacek 1985 .

4 Ž . Ž . Ž .See Ehrenberg and Milkovich 1987 , Ehrenberg 1990 , Ichniowski and Shaw 1993 .5 Ž . Ž . Ž .See Dickens and Katz 1987 , Gibbons and Katz 1992 , Groshen 1991 , Krueger and Summers

Ž . Ž .1988 , Thaler 1989 .

HIGH WAGE WORKERS 253

person and firm effects do not depend upon which effect is estimated first andthe disadvantage that it cannot impose orthogonality between the estimated

Ž .residual and the model effects a characteristic of the full least squares solution .The other conditional method, called ‘‘order dependent,’’ has the advantage ofimposing this orthogonality but the disadvantage of giving different resultsdepending upon which order is used to estimate the person and firm effects. Inparticular, the outcome of ‘‘persons first and firms second’’ would differ from‘‘firms first and persons second.’’ In all our estimated models, we find thatperson effects are statistically more important than firm effects in explainingcompensation and performance outcomes and that the two effects are not highlycorrelated. Using our consistent estimation method, we show empirically thatany method in which persons effects are estimated first, whether firm effects areestimated at the same step or after the person effects, performs better thanmethods in which person effects are estimated after firm effects.

We use our statistical decomposition of wage rates into person and firmeffects to address several classic questions in labor economics�the basis forinter-industry wage differentials, the source of the firm size-wage rate relation,the effect of seniority on wage rates, and the relation between pay structure,productivity, and profitability. Surprisingly, our French data give a clear answerto the first question. Virtually all of the inter-industry wage differential isexplained by the variation in average individual heterogeneity across sectors.Person effects, and not firm effects, form the basis for most of the inter-in-dustrial salary structure. A very large portion of the positive firm-size wage-raterelation is also due to person effects. The effect of seniority on wage rates isquite heterogeneous across firms; its estimated magnitude is very sensitive to theestimation technique. All our methods for estimating firm effects, includingheterogeneous seniority effects, perform well for large firms.

To study pay structure models, we aggregate individual components of com-pensation to the firm level. Then, we show that firms that hire high-wageworkers are more productive per worker, more capital intensive, more profes-sional-employment intensive, and more likely to survive. These same firms arenot more profitable, nor are they more skilled-labor intensive. Second, we show

Ž .that high-wage firms are more profitable, more productive per worker, possiblyŽ .more professional-employment intensive, and possibly more capital intensive.

Ž .High wage firms are unskilled labor intensive and possibly less likely to survive.The paper is organized as follows. In Section 2, we present a detailed

motivation of our statistical model in which we relate the different componentsof our statistical model to wage rate determination models used to studyinter-industry wage differentials, firm-size wage effects, the measurement ofopportunity wage rates, seniority-wage effects, and the economics of humanresource management. In Section 3 we lay out the full details of our statisticalmethods. We discuss the institutional features of the French labor market andour data sources in Section 4. In Section 5 we discuss our results. Finally, weconclude in Section 6.


2. HETEROGENEITY AND LABOR MARKETS

That labor market outcomes are extremely heterogeneous�observably equiv-alent individuals earn markedly different compensation and have markedlydifferent employment histories�is one of the enduring features of empiricalanalyses of labor markets in many countries. This heterogeneity has motivatedan enormous literature that attempts to isolate its sources and to identifysignificant market factors that are statistically related to employment outcomes,particularly earnings or compensation.6 One strand of this literature has focusedon the extent to which wage heterogeneity is related to permanent unmeasureddifferences among the individuals, what we label a person effect. Another strainof this literature has focused on the extent to which wage heterogeneity isrelated to permanent differences among the employers, what we label a firmeffect.

To put these different models in context, consider the following simple wageequation:

Ž . Ž .2.1 y �� x �� i t y i t x i JŽ i , t . i t

in which y is the logarithm of annual compensation of individual i�1, . . . , N ati tdate t�1, . . . T ; x is a vector of P time-varying exogenous characteristics ofi tindividual i; � is the pure person effect; � is the pure firm effect for thei JŽ i, t .

Ž Ž ..firm at which worker i is employed at date t denoted by J i, t , � is the grandymean of y , � is the grand mean of x , and � is the statistical residual.i t x i t i tAssume that a simple random sample of N individuals is observed for T years.7

Thus, � has the following properties:i t

� Ž . �E � � i , t , J i , t , x �0i t i t

and� Ž . Ž . �cov � , � � i , t , n , s, J i , t , J n , s , x , xit n s i t n s

� 2 for i�n and t�s,�� ½ 80 otherwise.

6 Ž . Ž . Ž . Ž .See, for example, Rosen 1986 , Willis 1986 , Becker 1993 , Juhn, Murphy, and Pierce 1993 ,Ž . Ž .Murphy and Welch 1992 , and Blau and Kahn 1996 .

7 The actual data are in the form of an unbalanced panel. For notational simplicity, however, wedescribe the motivation in terms of a balanced panel. Our complete model is described in the nextsection and the proofs for the unbalanced case are given in the Statistical Appendix.

8 One can always allow for a more complicated error structure for � ; however, as Abowd andi tŽ .Card 1989 show, except for measurement error, this residual exhibits trivial serial correlation in

American longitudinal data. Measurement error in the data studied by Abowd and Card, which doesexhibit significant serial correlation within individuals, is related to the structure of samples ofindividuals in which the individuals are the respondents. In this paper, we study data sampled at thelevel of the individual but reported by the employer; hence, respondent reporting error and othersources of measurement error in individual longitudinal data are not important problems. We will,therefore, maintain the covariance structure assumptions stated here for simplicity. When weconsider consistent estimation of � below, we allow for a general covariance structure on � fori teach i.


In matrix notation we have

Ž .2.2 y�X��D��F�� ,� Žwhere X is the N �P matrix of observable, time-varying characteristics in

. �deviations from the grand means , D is the N �N matrix of indicators forindividual i�1, . . . , N, F is the N� �mJ matrix of indicators for the firm effect

Ž . 9 �at which i works at date t J firms total , y is the N �1 vector of annualŽ .compensation data also in deviations from the grand mean , � is the con-

� Ž .formable vector of residuals, and N �NT. The parameters of equation 2.2are � , the P�1 vector of coefficients on the time-varying personal characteris-tics; � , the N�1 vector of individual effects;10 � , the mJ�1 vector of firmeffects; and the error variance, � 2.�

Ž . Ž .Equations 2.1 and 2.2 are interpreted as the conditional expectation ofannual compensation given information on the observable characteristics, thedate of observation, the identity of the individual, and the identity of theemploying firm. The discussion that follows clarifies the interpretation of classi-cal least squares estimates of the parameters � , � , and � when some of theseeffects are missing or are aggregated into linear combinations. The specification

Ž .in equation 2.2 is a simplification of the model used in our full analysis below,which we adopt in this section to clarify the discussion. All of the resultsdiscussed in this section are general and our Statistical Appendix containsproofs for the general case implemented in our data analyses. As the assump-

Ž . Ž .tions on the error process make clear, equations 2.1 and 2.2 impose theassumption of exogenous mobility. In particular, the design matrix for the firmeffects, F, is orthogonal to the error process � . Although endogenous mobility isclearly an important problem, we maintain the assumption of exogenous mobil-ity throughout this paper because we are interested in measuring and summariz-ing the role of personal and firm heterogeneity in the wage outcomes. Theextent to which such heterogeneity arises from endogenous mobility, or otherconsiderations, is the subject of future analyses.

Ž .Because many authors have estimated variations of 2.2 , but not the fullmodel, there is considerable ambiguity about the interpretation of variouscombinations of these parameters.11 Leaving aside the distinction between the

9 For simplicity in this section we treat the case m�1, so that the firm effect is a constant foreach firm. Later in the text we analyze more general firm effects.

10 Ž .The parameter � includes both the unobservable to the statistician individual effect and thecoefficients of the non-time-varying personal characteristics.

11 Since we began working on this paper; several working papers have appeared that use aŽ . Ž .specification similar to equation 2.2 . In particular, see Goux and Maurin forthcoming , who

Ž .calculate the exact least squares solution for the modal in equation 2.2 using French data with aŽ .much smaller sample of firms and persons than we use; Entorf, Gollac, and Kramarz forthcoming ,

who also compute the exact least squares solution using French data with fewer firms and personsŽ . Ž .than the present paper; and Belzil 1996 and Bingley and Westergard-Nielsen 1996 , who use˚

Ž .Danish data but do not compute the full least squares solution to equation 2.2 ; instead, theyŽ .assume orthogonal firm and person effects. Leonard and Van Audenrode 1996b use a specification

similar to the present one on Belgian data.


conditional and structural interpretation of the parameters, about which wehave nothing further to add, it is important to note that the omission or

Ž .aggregation of one or more of the effects in equation 2.2 can change themeaning of the other effects significantly. Variations in the set of conditioningeffects, which give rise to omitted-variable biases, are one source of confusionabout the interpretation of the statistical parameters. The use of different linear

Ž .combinations of the effects in equation 2.2 , which gives rise to aggregationbiases, is another source of differential interpretations for the parameters. We

Ž .investigate each of these variations in the parameterization of equation 2.2 inthe context of different problems in labor economics.

Ž .When the estimated version of equation 2.2 excludes the pure firm effectsŽ . �� , the estimated person effects, � , are the sum of the pure person effects, � ,and the employment-duration weighted average of the firm effects for the firmsin which the worker was employed, conditional on the individual time-varyingcharacteristics, X :

�1� ��Ž . Ž .2.3 � �� D M D D M F� ,X X

Ž � .�1 �where the notation M �I�A A A A for an arbitrary matrix A. Hence, if XAwere orthogonal to D and F, so that D�M D�D�D and D�M F�D�F, thenX Xthe difference between � � and � , which is just an omitted variable bias, wouldbe an N�1 vector consisting, for each individual i, of the employment-duration

� Ž . Ž .4weighted average of the firm effects � for j� J i, 1 , . . . , J i, T :j

T �JŽ i , t .�� .Ýi i Tt�1

The estimated coefficients on the time-varying characteristics in the case ofomitted firm effects, � � , are the sum of the parameters of the full conditionalexpectation, � , and the omitted variable bias that depends upon the conditionalcovariance of X and F, given D:

�1� �� Ž .� �� X M X X M F� .D D

Ž .Similarly, omitting the pure person effects � from the estimated version ofŽ . ��equation 2.2 gives estimates of the firm effects, � , that can be interpreted as

the sum of the pure firm effects, � , and the employment-duration weightedaverage of the person effects of all of the firm’s employees in the sample,conditional on the time-varying individual characteristics:

�1� ��Ž . Ž .2.4 � �� F M F F M D� .X X

Hence, if X were orthogonal to D and F, so that F�M F�F�F and F�M D�X XF�D, then the difference between � �� and � , again an omitted variable bias,would be a J�1 vector consisting, for each firm j, of the employment-duration


� Ž . 4weighted average of the person effects � for i� J i, t � j for some t :i

N T Ž Ž . .� 1 J i , t � ji�� Ý Ýj j Nji�1 t�1

where

N T

Ž Ž . .N � 1 J i , t � jÝ Ýji�1 t�1

Ž .and the function 1 A takes the value 1 when A is true and 0 otherwise. Theestimated coefficients on the time-varying characteristics in the case of omittedindividual effects, � �� , are the sum of the parameters of the full conditionalexpectation, � , and the omitted variable bias that depends upon the covarianceof X and D, given F:

�1� ��Ž . Ž .2.5 � �� X M X X M D� .F F

Ž .Almost all existing analyses of equations like 2.2 produce estimated effectsthat confound pure person and pure firm effects in a manner similar to thosepresented above. The possibility of identifying both person and firm effects thusallows us to reexamine many important topics in labor economics using esti-mates that properly allocate the statistical effects associated with persons andfirms.

2.1. Inter-industry Wage Differentials

Consider now the analysis of inter-industry wage differentials as done byŽ . Ž .Dickens and Katz 1987 , Krueger and Summers 1987, 1988 , Murphy and

Ž . Ž .Topel 1987 , Gibbons and Katz 1992 , and many others. The principal findingof this literature has been that inter-industrial wage differentials cannot beexplained by measured person or firm characteristics. There is continuingcontroversy regarding the extent to which these differentials are explained byunmeasured person effects, with Krueger and Summers claiming that they are

Ž .not Gibbons and Katz concurring , Murphy and Topel claiming that unmea-sured person effects are the primary explanation, and Dickens and Katz not ableto address the issue. As we make clear in this section, the ability to estimateboth person- and firm-level heterogeneity will permit us to substantially resolvethis question in our data analysis�in favor of the person-effect explanation, asit turns out.

To standardize notation and parameter interpretation, define the pure inter-industry wage differential, conditional on the same information as in equationsŽ . Ž .2.1 and 2.2 , as for some industry classification k�1, . . . , K. Industry is akcharacteristic of the firm; thus, our definition of the pure industry effect issimply the correct aggregation of the pure firm effects within the industry. Weselect the definition of an industry effect as the one that corresponds to putting


Ž .industry indicator variables in equation 2.2 and, then, defining what is left ofthe pure firm effect as a deviation from the industry effects. Hence, can bekrepresented as an employment-duration weighted average of the firm effectswithin the industry classification k:

N T Ž Ž Ž .. .1 K J i , t �k �JŽ i , t . � ,Ý Ýk Nki�1 t�1

where

J

Ž Ž . .N � 1 K j �k NÝk jj�1

Ž .and the function K j denotes the industry classification of firm j. If we insertthis pure industry effect, the appropriate aggregate of the firm effects, into

Ž .equation 2.1 , then the equation becomes

Ž .y �x �� i t i t i KŽJŽ i , t .. JŽ i , t . KŽJŽ i , t .. i t

Ž .or, in matrix notation as in equation 2.2 ,

Ž . Ž .2.6 y�X��D��FA� F��FA ��

where the matrix A, J�K, classifies each of the J firms into one of the KŽ .industries; that is, a �1 if, and only if, K j �k. The parameter vector ,jk

K�1, may be interpreted as the following weighted average of the pure firmeffects:

�1� � � �Ž .� A F FA A F F� ,

Ž .and the effect F��FA may be re-expressed as M F� . Thus, the aggrega-FAtion of J firm effects into K industry effects, weighted so as to be representative

Ž .of individuals, can be accomplished directly by estimation of equation 2.6 . OnlyŽ � .rank F M F firm effects can be separately identified; however, there isFA

neither an omitted variable nor an aggregation bias in the classical least squaresŽ . Ž .estimates of 2.6 . To be perfectly clear, equation 2.6 decomposes F� into two

orthogonal components: the industry effects FA , and what is left of the firmeffects after removing the industry effect, M F� .FA

Ž . Ž .Authors like Dickens and Katz 1987 , Krueger and Summers 1987, 1988 ,Ž . Ž .Murphy and Topel 1987 , and Gibbons and Katz 1992 do not have informa-

tion identifying the employing firm, even when they do have longitudinal data.12

Estimates of industry effects, � , that are computed on the basis of an equationthat excludes the remaining firm effects, M F� , are equal to the pure industryFAeffect, , plus an omitted variable bias that can be expressed as a function of theconditional variance of the industry effects, FA, given the time-varying charac-

12 Ž .Krueger and Summers 1988, Table V , for example.


teristics, X, and the person effects, D,

�1� � � �� Ž . �� A F M FA A F M M F� ,� D X � � D X � FA

which simplifies to � � if, and only if, the industry effects, FA, are orthogo-nal to the subspace M F, given D and X, which is generally not true evenFAthough FA and M F are orthogonal by construction.13 Thus, it is not possibleFAto estimate pure inter-industry wage differentials consistently, conditional ontime-varying personal characteristics and unobservable non-time-varying per-sonal characteristics, without identifying information on the underlying firmsunless this conditional orthogonality condition holds. Similarly, estimates of thecoefficients of the time-varying personal characteristics, � � , are equal to thetrue coefficients of the conditional expectation, � , plus an omitted variable biasthat depends upon the conditional covariance between these characteristics, X,and the residual subspace of the firm effects, M F, given D:FA

�1� �� Ž .� �� X M X X M M F� ,� D FA � � D FA � FA

which, once again, simplifies to � � �� if, and only if, the time-varying personalcharacteristics, X, are orthogonal to the subspace M F, given D and FA,FAwhich is also not generally true. Thus, it is not possible to estimate thecoefficients on time-varying personal characteristics consistently, conditional onindustry effects and unobservable non-time-varying personal characteristics,without identifying information on the underlying firms unless this secondconditional orthogonality condition holds.

Ž .When the estimation of equation 2.6 excludes both person and firm effects,the estimated industry effect, �� , equals the pure industry effect, , plus thekemployment-duration weighted average residual firm effect inside the industry,given X, and the employment-duration weighted average person effect insidethe industry, given the time-varying personal characteristics X :

�1� � � �� Ž . Ž . �� A F M FA A F M M F��D� ,X X FA

which can be restated as

�1 �1� � � � � � � ��Ž . Ž . Ž .2.7 � A F M FA A F M F�� A F M FA A F M D� .X X X X

Hence, if industry effects, FA, were orthogonal to time-varying personal charac-teristics, X, and to non-time-varying personal heterogeneity, D, so thatA�FM FA�A�F�FA, A�F�M F�A�F�F, and A�F�M D�A�F�D, the biased in-X X Xter-industry wage differentials, �� , would simply equal the pure inter-industrywage differentials, , plus the employment-duration-weighted, industry-average

13 � � �M is the matrix M with Z� D X and is not equal to the matrix M .� D X � Z D X


Ž � � .�1 � �pure person effect, A F FA A F D� , or

N T � Ž Ž .. �1 K J i , t �k � i�� ,Ý Ýk k Nki�1 t�1

� Ž Ž .. �where N �Ý 1 K J i, t �k .k i, tThus, previous analyses that exclude person effects confound the pure inter-

industry wage differential with an average of the person effects found in theindustry, given the measured personal characteristics, X. To anticipate our

Ž .results, we use equation 2.7 together with our estimated pure person effects, � ,and our estimated pure firm effects, � , to determine what proportion of theestimated inter-industry wage differentials �� is explained by person effectsversus firm effects. We show that the pure inter-industry wage differential, ,which we interpret, as in this section, as the part due to pure firm effects, ismuch less important than the contribution of the industry average person effectto ��.

2.2. Firm Effects without Personal Heterogeneity

There is a complementary line of research that attempts to explain hetero-Ž .geneity in wage rates by using firm effects, for example Groshen 1991, 1996 ,

Ž . Ž .Davis and Haltiwanger 1996 , Entorf and Kramarz 1997, forthcoming andŽ .Kramarz, Lollivier, and Pele 1996 . The principal finding in these studies has´

been that firm effects are substantially more important than measured personalcharacteristics in explaining wage variation, even when the measured personalcharacteristics include detailed occupational effects, which are typically inter-preted as a proxy for our pure person effects, � . An additional conclusion is thatthe effects of measured personal characteristics, � , are not very sensitive to theinclusion of firm effects. None of the studies in this strain of the wage-de-termination literature includes both pure person and pure firm effects, as

Ž . Ž .defined in equation 2.1 or 2.2 above.Ž . ��In our notation, studies like Groshen 1991 estimate � , from equation

Ž . �� Ž .2.4 , and � , from equation 2.5 . The size of the bias arising from theomission of person effects is, of course, an empirical matter; however, again toanticipate our results, it turns out to be substantial. Most of the estimated firmeffect, � �� , in these studies is due to the employment-duration weighted

Ž � .�1 �average of the pure individual effects conditional on X, F M F F M D� ,X Xand not to the pure firm effect, � . Furthermore, the bias in the estimated effects

�� Ž � .�1 �of time-varying personal characteristics, � �� X M X X M D� , dueF Fto the omission of pure individual effects, is also large.

2.3. Firm-Size Wage Effects

The repeated finding of a positive relation between the size of the employingfirm and wage rates, even after controlling for a wealth of individual variables


Ž Ž ..see Brown and Medoff 1989 , has generated many alternative interpretations.Some explanations rely on efficiency wage considerations�monitoring beingmore difficult in larger firms�or, more generally, upon firm-specific compensa-tion policies.14 Others rely on the assumed existence of unobserved workercharacteristics, compensated by the firms, that only larger firms would be able tospot because of better hiring practices.15 The estimated firm-size effect on wagerates can be related to what we call pure firm effects as well as to the averageperson effect within the firm. Using our notation, a firm-size effect, , can bemodeled using a matrix S, J�R, that maps the size of firm j into R linearly

Žindependent functions of its size polynomials in the logarithm or size intervals,.for example . Following the same methods that we used to decompose the

Ž .inter-industry wage differential, we express the wage equation 2.2 as:

Ž .2.8 y�X��D��FS�M F�� ;FS

so that the pure firm-size effects are related to the underlying pure firm effectsby the equation

�1� � � �Ž .� S F FS S F F� .

Once again, we stress that firm size is a characteristic of the employer; thus, afirm-size effect is simply an aggregation of the pure firm effects and can beanalyzed using the same tools that we used for the inter-industry wage differen-tial. Therefore, all of the bias formulas derived for the inter-industry wagedifferential apply to the problem of estimating the firm-size effects in the

Ž .presence or absence of the various effects in equation 2.8 . In particular, whenthe firm-size effects are estimated in the presence of measured time-varyingpersonal characteristics, X, and person effects, D, but omitting the remainingfirm effects, M F, the resulting estimated firm-size effects, � , as in BrownFS

Ž .and Medoff 1989, Table 2 take the form

�1� � � �� Ž . �� S F M FS S F M M F�� D X � � D X � FS

with a similar equation, which we do not state explicitly, for the bias in theŽ .estimation of the parameters � in equation 2.8 . The firm-size effects estimated

in the absence of firm effects, � , are equal to the pure firm-size effects, , if,and only if, firm size, FS, is orthogonal to the residual subspace of firm effects,M F, given time-varying personal characteristics, X, and person effects, D. AsFSin the case of industry effects, we note that this conditional orthogonality doesnot follow from the fact that FS and M F are orthogonal by construction.FSHence, the bias � � is not generally zero.

Most studies of the firm-size wage effect do not condition on person effects,D. Consequently, the estimated parameter vector associated with the firm-size

14 Ž .See Bulow and Summers 1976 , for example.15 Ž .See Weiss and Landau 1984 , for example.


�� Ž .effect in those studies, in our notation , can be represented as

�1 �1� � � � � � � ��Ž . Ž . Ž .2.9 � S F M FS S F M F�� S F M FS S F M D� ,X X X X

which we interpret as the sum of the firm-size, employment-weighted averagefirm effect and the similarly-weighted average person effect, conditional onpersonal characteristics, X. To anticipate our results, again, we use the decom-

Ž .position displayed in equation 2.9 to explain the relation between firm size andthe firm-size class average person and firm effects in our data, conditional onother firm-level and personal variables. The relation between firm size and thesecomponents of wage outcomes is, as Brown and Medoff hypothesized, impor-tantly related to both pure firm heterogeneity in compensation, � , and pureindividual heterogeneity, � .

2.4. Measurement of the Internal and External Wage

Virtually all economic models of labor market outcomes require an estimateof the opportunity cost of the worker’s time. In simple, classical equilibriummodels without unmeasured person or firm heterogeneity, this generally corre-sponds to the measured wage rate. In models of wage determination such as

16 Žquasi-rent splitting or imperfect information efficiency wage and agency. 17 Ž .models , unmeasured statistical heterogeneity person or firm breaks the

direct link between the observed wage rate and the opportunity cost of time.Moreover, such models usually make an explicit distinction between the com-pensation received and the wage rate available in the employee’s next best

Ž .alternative employment. The statistical model in equation 2.1 , while notderived from an explicit labor market model, contains all the observable ele-

16 In the collective bargaining and wage determination literature, this problem has a longŽ Ž . Ž .theoretical history see Leontief 1946 , MacDonald and Solow 1981 , and most recently Manning

Ž .. Ž . Ž .1987 . Many empirical implementations, including Brown and Ashenfelter 1986 , Card 1986 ,Ž . Ž . Ž .MaCurdy and Pencavel 1986 , Abowd 1989 , Christofides and Oswald 1991, 1992 , Nickell and

Ž . Ž . Ž .Wadhwani 1991 , Abowd and Lemieux 1993 , and Blanchflower, Oswald, and Sanfey 1996 , usemacro-economic wage series or sectoral wage series to represent the opportunity cost of time for theunionized workers. This technique fails to capture important variation in the average personal

Ž .heterogeneity of the employees of different firms. See Abowd and Kramarz 1993 and Abowd andŽ .Allain 1996 for empirical models that permit unobserved heterogeneity in the opportunity cost of

time.17 Ž .For agency models, the theory is summarized in Hart and Holmstrom 1987 and Sappington¨

Ž . Ž . Ž . Ž .1991 . Some empirical implementations include Lazear 1979 , Hutchens 1987 , Abowd 1990 ,Ž . Ž . Ž .Gibbons and Murphy 1990, 1992 , Jensen and Murphy 1990 , Kahn and Sherer 1990 , Leonard

Ž . Ž . Ž . Ž .1990 , Cahuc and Dormont 1997 , Cahuc and Kramarz 1997 , Kramarz and Rey 1995 , andŽ .Leonard and Van Audenrode 1996a , all of which require an empirical proxy for the external wage

rate in order to identify a component of compensation that is related to performance. See also theŽ .summary in Ehrenberg and Milkovich 1987 . For efficiency wage models, the theory is summarized

Ž . Ž .in Shapiro and Stiglitz 1984 , for the dual labor market version see Bulow and Summers 1976 andŽ . Ž .Cain 1976 . Again, empirical models like Dickens and Lang 1985 require a measure of the

opportunity cost in the low-wage sector. The measures used do not allow for unobserved personalheterogeneity between the low and high observed wage groups.


ments from which nonclassical labor market models derive their empiricalcontent. Indeed, the simplest definition of the components of the external and

Ž .internal wage rate based on a structural model leading to equation 2.1 is givenby the following model:

Ž .2.10 y �x ��i t i t i t

� 4where x , � follows a general stochastic process for i�1, . . . , N and t�1, . . . Tit i twith

Ž . �� 4 � 4 Ž . Ž .�2.11 E x , � x , � � i , n , s, t , J i , t , J n , s �0i i i t n s n s

Ž . Ž .iff i�n or J i , t �J n , s .Then,

� � � �� E x �� i �E x ��i i t i t i t i t

and

� Ž . � � �� E x �� J i , t � j �E x �� .j i t i t i t i t

Ž . Ž .The model in equation 2.10 , together with the assumption 2.11 , simplyformalizes the conditions under which we can use our maintained assumption of

Ž .exogenous mobility to apply a structural interpretation to equation 2.1 .

2.5. Analysis of the Seniority-Wage Rate Relation

In the growing literature on the effects of seniority on wage rates, mostauthors assume that the relevant coefficient is homogeneous across firms.18

Ironically, the first uses of the seniority-wage relation to test economic theoriesŽ Ž . Ž ..Lazear 1979 and Hutchens 1987 do not make this assumption. Further-

Ž .more, Margolis 1996 has shown, using estimated seniority effects related tothose presented in the present paper, that heterogeneity in the returns toseniority is a significant empirical phenomenon and that one’s interpretation ofthe average effect of seniority on wage rates is affected by whether or not themodel allows for the heterogeneity. The seniority-wage relation is a firm-specifictime-varying effect. Thus, the statistical techniques developed in this paper canbe used to model and estimate this effect. We extend the analysis in MargolisŽ .1996 by including a heterogeneous seniority effect in several statistical models.We provide consistent estimates of this effect within firms using assumptions

Ž .that are comparable to Topel’s 1991 assumptions. We compare these resultswith other estimation techniques that assume heterogeneous or homogeneousseniority effects. Furthermore, we provide direct evidence on the extent to whichthe between-firm variability in returns to seniority is related to the between-firm

Žvariability in initial pay. Several models of lifetime incentive contracts BeckerŽ . Ž ..and Stigler 1974 , Lazear 1979 predict a negative relation, which our statistics

support.

18 Ž . Ž . Ž . Ž .See Abraham and Farber 1987 , Altonji and Shakotko 1987 , Brown 1989 , and Topel 1991 .


2.6. Human Resource Management Policies

In the emerging literature on the economics of human resource managementŽ Ž . Ž ..policies see Ehrenberg 1990 and Lazear 1998 , economists and other organi-

zation specialists have argued that a firm’s personnel practices, particularly thedesign of its compensation policy, are directly related to the performance of thefirm. These ideas, which we can consider formally in the context of statistical

Ž .models like equation 2.1 , take us back directly to the questions we posed in theintroduction. We will measure the opportunity wage of our workers using ourestimate of the person-specific heterogeneity in compensation. Thus, at the firmlevel, the presence of high-wage workers is measured by the average of theperson-specific heterogeneity component of pay. The extent to which the firm,through its hiring practices, selects employees who are, on average, better orworse paid than observably-equivalent employees in other firms is, then, directlyrelated to other firm-level outcomes. Again at the firm level, the presence of ahigh-wage policy will be measured by the firm-specific component of compensa-tion. The extent to which a firm, through its compensation policy, attempts topay above or below the prevailing market is, then, directly related to otherfirm-level outcomes. Firm outcomes of interest include the average productivity

Ž .of labor, sales per employee as measures of productivity , and the operatingŽ .income per unit of capital as a measure of profitability . Existing empirical

studies have attempted to relate similar profitability or productivity measures tospecific components of the firm’s human resource management practices.19

Because we have a large, representative sample of firms and easily-understoodmeasures of the firms’ compensation policies, we are able to supply very directstatistical evidence on the importance of these human resource managementpractices on the performance and the structure of the firm.

3. STATISTICAL MODEL

3.1. Specification of the General Model

Ž .Consider, again, our full model as described in equation 2.2 . To make ouranalysis general enough for the data we use, we note that the rows of y, X, D,and F are arranged in the order i�1, . . . , N and, within each i, t�n , . . . , n ,i1 iTi

where T is the total number of years of data available for individual i and thei

indices n , . . . , n indicate the year corresponding to the first observation oni1 iTi

individual i through the last observation on that individual, respectively. Thus

19 Ž .See almost all of the studies in Ehrenberg 1990 but, in particular, Abowd, Hannon, andŽ . Ž . Ž .Milkovich 1990 , Kahn and Sherer 1990 , and Leonard 1990 . Other studies include Weiss and

Ž . Ž . Ž .Landau 1984 , Ehrenberg and Milkovich 1987 , Cahuc and Dormont 1997 , Ichniowski and ShawŽ . Ž . Ž .1993 , Cahuc and Kramarz 1997 , Abowd, Kramarz, and Moreau 1996 , and Leonard and Van

Ž .Audenrode 1996b .


the vector y is organized as

y1, n11

y1, n1T 1

Ž .3.1 y� ; yN , nN1

yN , nN TN

X, D, F, and � are organized conformably; and � , the parameter vectorassociated with the firm effects, is mJ�1 with m�1. To simplify the notationwe will refer to a typical element of y as y and a typical element of X, or anyi, t

Ž .similarly organized matrix, as x where the pair i, t denotes the row index.Ž i, t ., jIn all of our statistical models, we decompose the person effect, � , into a parti

that is related to non-time-varying personal characteristics, u , and a part that isinot observable to the statistician, � . We use the orthogonal decomposition of �i idefined byŽ .3.2 � �� u �i i i

where u is a vector of non-time-varying measurable personal characteristics, �i iis the person-specific intercept, and � is the vector of coefficients. We also usethe following decompositions of � . The first of these defines a firm effect withjm�2,Ž .3.3 � �� s ,j j j i t

Ž .where s denotes individual i’s seniority in firm j�J i, t in year t, � denotesi t jthe firm-specific intercept, and � is the firm-specific seniority coefficient. Thejsecond decomposition of � defines a firm effect with m�3:j

Ž . Ž .3.4 � �� s �� T s �10 ,j j j i t 2 j 1 i t

Ž . Ž . kwhere T x �0 when x�0 and T x �x when x�0, and � measures thek k 2 jchange in the firm-specific seniority coefficient that occurs after 10 years of

Ž .seniority. In matrix form equation 3.3 decomposes F� asŽ .3.5 F��F ��F �0 1

where F is the N� �J design matrix associated with the vector of firm specific0intercepts, F is the N� �J matrix whose columns consist of the direct product1of the columns of F and an N� �1 vector whose elements are s , � is the0 i tJ�1 vector of firm-specific intercepts and � is the J�1 vector of firm-specific

Ž .seniority coefficients. In matrix notation, equation 3.4 decomposes the firmeffect asŽ .3.6 F��F ��F ��F �0 1 2 2

where F is the N� �J matrix whose columns consist of the direct product of2� Ž .the columns of F and an N �1 vector whose elements are T s �10 , and �0 1 i t 2


is the J�1 vector of firm-specific changes in the seniority coefficient after 10years of seniority.

For completeness, we also note that the derivation of some of our specifica-Ž 2 .tion tests requires the assumption that �N 0, � I . This completes the�

notation used in the general specification of our statistical model.

3.2. Identification of Parameters in the General Model

We now consider basic issues in the identification of the parameters of ourŽ .model. Although equation 2.2 is just a classical linear regression model, the

� � Žfull design matrix X D F has high column dimension N�1,000,000 and.J�50,000, estimable . The cross-product matrix

� � �X X X D X F� � �D X D D D F� � �F X F D F F

is patterned in the elements D�D and F�F; however, projecting onto thecolumns D leaves a 100,000�100,000 unpatterned, nonsparse matrix to invert

Ž .when m�2 the linear seniority effect case because workers move betweenfirms. Indeed, mobility is a necessary condition if one wants to separatelyidentify person effects, � , and firm effects, � , in the general model. Similarly,projecting onto the columns of F leaves a 1,000,000�1,000,000 unpatterned,nonsparse matrix to invert. Clearly, the usual computational methods for least

� � � � ��squares estimation of the parameter vector � � � are not feasible. Hence,because one cannot compute the unconstrained least squares estimates for the

Ž .model 2.2 , we propose several different estimators that attempt to preserve asmuch of the general structure of the problem as is computationally possible.

Although we do not discuss the origin of our data until Section 4, one aspectof the data, inter-firm mobility, is so critical to the estimation and interpretationof our analyses that we present a summary now. Regardless of the computa-tional approach used, between-employer mobility of the individuals is essentialfor the identification of our statistical model. Table I examines the pattern ofinter-employer movements among all sample individuals. The rows of Table Icorrespond to the number of years a person is in the sample. The columns, with

Ž .the exception of column la , correspond to the number of employers theŽindividual had. An individual contributes to only one cell again, excepting

Ž ..column la . Notice that 59.4% of the individuals in the sample never changeŽ Ž .. 20employers column 1 . Approximately one-fifth of the single employer indi-

20 Ž .Notice that the cell 1, 1 contains 318,627 individuals who appear in the sample during a singleyear. Some of these individuals may represent coding errors in the person identifier; however, it isnot possible to correct these errors.


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Žviduals worked in firms with no movers while four-fifths 47.9% of the overallŽ ..sample, column 1a worked in firms that, at one time or another, employed a

person who changed employer. Thus, 88.5% of the sample individuals contributeto the estimation of firm-effects. It is also interesting to notice the pattern of

Ž Ž . Ž ..employer spells among the movers columns 2 � 10 . The second line of eachcell shows the most frequent configuration of employer spells for individuals inthat cell. In almost every case, short spells precede longer spells, indicating that

Ž Ž .mobility is greater earlier in the career as Topel and Ward 1992 found for.American men . It seems clear from Table I that the data should allow us to

separate the individual effect from the firm effect.

3.3. Identification and Consistent Estimation of � and � j

In this subsection we show how to obtain consistent estimates of � and � jusing the within-individual-firm differences of the data. This method provides uswith our most robust statistical method in the sense that we use no additional

Ž .statistical assumptions beyond those specified in equation 2.1 and definitionŽ .3.3 . Consider the first differences:

Ž . Ž . Ž .3.7 y �y � x �x �� s �s �� in in in in JŽ i , n . in in in init i t�1 i t i t�1 i t i t i t�1 i t i t�1

Ž . Ž .for all observations for which J i, n �J i, n , which we represent in matrixi t i t�1form as

˜Ž .3.8 � y��X ��F��

� � � � �˜where � y is N �1, �X is N �P, F is N �J, �� is N �1, and N is equalŽ .to the number of i, t combinations in the sample that satisfy the condition

˜Ž . Ž .J i, n �J i, n . The matrix F contains the rows of F that correspond toi t i t�1 1Ž . Ž . Ž .the person-years i, t for which the condition J i, n �J i, n is satisfiedi t i t�1

minus the immediately preceding row. Then,�1� �˜Ž . Ž .3.9 �� X M �X �X M � y˜ ˜F F

and�1

� �˜ ˜ ˜ ˜Ž . Ž .3.10 �� F F F � y��X � .˜ Ž .˜� �A consistent estimate of V � is given by

��1 �1� � �˜ ˜Ž . Ž .V � � �X M �X �X M � M �X �X M �X˜ ˜ ˜ ˜Ž .F F F F

where

˜ � �� 0 01

˜ � �0 � �� 02��

˜ � �0 0 � ��N


and

� � � � �2�� in in in in in2 2 3 2 Ti� � � � �2�� in in in in in3 2 3 3 T˜ i� �� .i

� � � � �2�� in in in in inT 2 T 3 Ti i i

Ž .It is understood that only the rows of �� that satisfy the condition J i, n �i t� �� 21˜Ž .J i, n are used in the calculation of � , which is therefore N � N .i t�1Ž . Ž .Notice that, given our assumptions, the resulting estimators 3.9 and 3.10

are also unbiased. Our consistent method is not unique but, it has the advantagethat the sample on which the estimation is performed includes both workerswho remained in the same firm at all dates as well as workers who movedbetween firms at some point in time during our analysis period. Even for thesemobile individuals, all first-differences for which the date t firm differs from thedate t�1 firm are not included in the estimating sample. Hence, our consistent

Ž .method is inefficient in the context of the specification of equation 2.2 . Inaddition to this inefficiency, we also note that our consistent method cannot beused to identify separately the firm intercept, �, and the person effect, � . Thisresults from the restriction of our analysis to a sample based on all observations

Ž . Ž .for which J i, n �J i, n . Any method that allows separate identification ofi t i t�1the two effects must include in some form the remaining observations. Hence,we turn now to other methods more appropriate to this purpose.

3.4. Conditional Estimation Methods

In this section we provide statistical models for estimating all of the effects inŽ .equation 2.2 using a class of estimators we call conditional methods because of

their relation to standard linear model computational techniques and because oftheir origins in the panel data literature on person-effect models.22 Our purposein developing these methods is to provide estimators that are as similar aspossible to the full least squares solution but that are computationally tractable.The basic idea is also simple. Since we cannot compute the full least squaressolution, we will have to impose some ancillary orthogonality assumptions inorder to proceed. We use information in the data in the form of higher orderinteractions between observable characteristics, person identity and firm iden-

21 ˜� �The formula for the consistent estimator of V � clearly allows for arbitrary correlation of theresiduals � over t for each i. Hence, our consistent estimator is unchanged if we permit ani tarbitrary time-series model for � .i t

22 Ž .The reader familiar with the analysis of variance as considered in, for example, Scheffe 1959´Ž .and Searle et al. 1992 , will notice that our conditional methods can also be derived as analysis of

covariance models in which the data are adjusted to remove certain effects, our conditioningvariables, before the conventional analysis of covariance formulas are applied to the model.


Ž .tity, which are excluded by hypothesis from equation 2.2 , to proxy for thecorrelation between X, D, and F. Then, we impose conditional orthogonality,given these higher order interactions. Since we have a consistent, but inefficient,estimator of some of the effects, we will use that estimator to assess the qualityof our conditional estimation methods when we consider formal specificationchecks. We will, thus, have some formal and some informal methods forcomparing a variety of estimators, none of which is the full least squares

Ž .solution for estimating the parameters of equation 2.2 .Consider a matrix of variables Z, N� �Q, which depends upon Q functions

of the information in X, D, and F. Using conditional methods we calculate theŽ .least squares estimates of equation 2.2 under different maintained hypotheses

about the conditional orthogonality of X, D, and F, given Z. The first of thesehypotheses imposes that the effects X and D be orthogonal to the projection ofF onto the null space of Z. Under this hypothesis the basic equation can berestated as

Ž .3.11 y�X��D��Z��M F��Z

Ž � .�1 �where the auxiliary parameter �� Z Z Z F� . The assumption of conditionalorthogonality between X and F, given Z, and between D and F, given Z,implies that

Ž . �3.12 X M F�0Z

and

Ž . �3.13 D M F�0.Z

Hence, the conventional least squares formula for the estimator of the original� � � � � �parameters, � � � , and the auxiliary parameters, �, is

�� ˆ X X X D X Z X M F� X yZ� � � � �D X D D D Z D M F D yˆ Z�Ž .3.14 � �� Z yZ X Z D Z Z Z M F� Z

�� F M yF M X F M D F M Z F M Fˆ Z� Z Z Z Z

� �� 23where the notation denotes a generalized inverse. Since the elements� � � Ž . Ž .X M F, D M F, and Z M F are zero, either by hypotheses 3.12 and 3.13 orZ Z Z

Ž .by construction, the formula 3.14 can be restated as��1� � �ˆ X y� X X X D X Z�� Ž . D y3.15 � D X D D D Z�

� � � �Z X Z D Z Z Z y�

and�� ˆŽ . Ž .3.16 �� F M F F M y.Z Z

23 Ž � .The use of a g-inverse is required because F M F is rank mJ�1�Q.Z


As we demonstrated in Section 3.3, certain parameters in our model can beestimated consistently without the use of ancillary assumptions like equationsŽ . Ž .3.12 and 3.13 . Consistent estimation of other parameters requires some extrahypotheses. If the conditional methods work well, then the conditional estimatesof � and � should not be too far from the estimates produced by the consistentj

method. This insight is the basis for the specification checks that we derivebelow.

3.4.1. Order-independent estimation

Ž .Our first method for the computation of the solution to equations 3.15 andŽ .3.16 can be accomplished in two steps, which can be performed in either order,hence our designation of this method as ‘‘order independent.’’ In the first step,

Ž .called the within-D step, the parameters in equation 3.15 are estimated byconventional longitudinal methods in which X and Z are projected on D toproduce the estimates of � and � given by

�1� � �X M X X M Z X M y� D D DŽ .3.17 � ,� � �Z M X Z M Z Z M yˆ D D D�

which are usually called the ‘‘within-person’’ estimators of these parameters.The associated estimator of � is

�1� �ˆ ˆ ˆŽ . Ž .3.18 �� D D D y�X��Z� .Ž .The second step in the computation of the complete set of order-independent,

Ž .conditional least squares estimates for equation 3.11 , called the within-F step,Ž .requires the solution of equation 3.16 . This is accomplished by computing the

least squares estimates of the parameters in the regression of y on F and Zjointly:

Ž .3.19 y�F��Z�� ,

Ž .where � is the same parameter vector that appears in equation 2.2 , � is aŽ 2 .Q�1 vector of auxiliary parameters, and N 0, � I because of the condi-

Ž . Ž .tional orthogonality conditions imposed in equations 3.12 and 3.13 . Computa-ˆtion of � is accomplished in two steps that are directly analogous to the method

Ž . Ž .used in equations 3.17 and 3.18 . First, compute � by the within-F estimatorˆ

�1� �Ž . Ž .3.20 �� Z M Z Z M y.ˆ F F

ˆThen, compute � with the estimator

�1�ˆŽ . Ž . Ž .3.21 �� F F F y�Z� .ˆ


Ž . Ž .The proof that the formulas 3.16 and 3.21 are equivalent follows. First, notethat

��1 �1 �1� � � � � � �Ž . Ž . Ž . Ž .�� Z Z � Z Z Z F F M F F Z Z Z Z yˆ Z

��1� � � �Ž . Ž .� Z Z Z F F M F F yZ

by direct application of the partitioned inverse formula to the full least squaresŽ .solution to equation 3.19 . Hence,

� �� Ž . Ž . Ž .3.22 y�Z�� M �P F F M F F P �P F F M F F yˆ Z Z Z Z Z Z

Ž . Ž .where P �I�M . Substituting equation 3.22 into equation 3.21 yieldsZ Z

�1� �ˆ Ž . Ž .�� F F F y�Z��1 �1� � � � � �Ž . Ž . Ž .� F F F M y� F F F P F F M F F P yZ Z Z Z

��1� � � �Ž . Ž .� F F F P F F M F F yZ Z

��1 �1� � � � � �Ž . Ž . Ž . Ž .� F F F M F � F F F P F F M F F M yZ Z Z Z

�� Ž .� F M F F M y. Q.E.D.Z Z

In some applications, the matrix F is just F , the design matrix for a single0Ž . Ž . Ž .firm-specific effect m�1 , and the computation of equations 3.20 and 3.21

can be accomplished by conventional formulas in which the values of y and Zare deviated from within-firm means in order to compute � . In our estimationûsing this conditional model we let m�2 in order to capture a firm-specific

Ž .intercept and seniority slope according to equation 3.5 . The within-F stepregression becomes

Ž .3.23 y�F ��F ��Z�� .0 1

ˆŽ .In estimation of the firm effects by equation 3.23 , the computation of � , �,ând � is more complex than for the case in which F�F . These complexitiesˆ 0are described in the Statistical Appendix.

The estimation of the correct covariance matrix for the combined within-Dand within-F estimator requires calculation of the correct residual for the full

Ž .model in 2.2 ,

ˆ ˆ ˆ ˆ�� y�X��D��Z��M F� .ˆ ž /Z

The computation of this residual is not straightforward. The first part of theresidual is computed at the within-D step as

�1� ˆ ˆ ˆ� � y�X��D��Z� .ˆ ž /The second part of the residual is computed at the end of the within-F step as

�2� ˆ ˆ ˜� �M F��F��Z�ˆ Z


˜ � �1 � ˆŽ .where �� Z Z Z F� . Finally,

�� 1� �� 2� .ˆ ˆ ˆThe standard analysis of variance estimator for the variance of the residual � isgiven by

�

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆy�X��D��Z��M F� y�X��D��Z��M F�ž / ž /Z Z2Ž .3.24 � �� Ž .N �P�N�Q� mJ�1�Q

where we note explicitly that the estimation of the mJ firm effects uses onlymJ�Q�1 degrees of freedom and that the Q degrees of freedom missing from

ˆthe firm effects have been used to estimate �. The proof follows:�1 �1� � � �Ž . Ž .�� I�W W W W �M F F M F F M ��M �ˆ Z Z Z �W M F �Z

� �where W� X D Z . Under the maintained orthogonality conditions in equa-Ž . Ž .tions 3.12 and 3.13 , the quadratic form

� ��2� .N *2��

and�1 �1� � � � � � � �Ž . Ž .� � � � � W W W W � � M F F M F F M �ˆˆ Z Z ZŽ .3.25 � � �2 2 2 2� � � ��

Ž � .�1 � Ž � .�1 �Since W W W W M F F M F F M �0, the last two quadratic forms onZ Z ZŽ . 2the right-hand side of equation 3.25 are independent � random variables

� Ž � .�1 � � � Ž � .�1 � �with rank W W W W �P�N�Q and rank M F F M F F M �mJZ Z Z�Q�1 degrees of freedom, respectively. Thus,

� ��ˆˆ2

�� .N �P�N�m J�12��

The error degrees of freedom for the complete model is, thus, N� �P�N�mJ�1, so that the dimensionality of the auxiliary parameter vector � does not

Ž .affect the goodness of fit of the model in equation 3.11 .

3.4.2. Order-dependent estimation

The order-dependent method, conditional on Z, means that the estimation ofcertain effects is performed before others; that is, that the residuals from thefirst step are used to compute the estimates of the second step. The result isorder-dependent because estimating person effects before firm effects is not thesame as estimating firm effects before person effects. We describe in detail theorder-dependent: persons first method. We comment only briefly on the order-dependent: firms first method, because the analogous formulas are straightfor-ward.


The first step of our order-dependent: persons first method uses the sameconditional estimation equations that were described above for the order-inde-pendent method to estimate the coefficients of the time-varying observablevariables, � , the person effects, � , and the conditioning effects, �, the coeffi-

Ž .cient of the variables Z. This is done according to equation 3.15 . Hence, theŽ . Ž .estimated coefficients are given by equations 3.17 and 3.18 .

In the second step of the order-dependent: persons first method, we estimateŽ . Ž . 24the firm effects using equation 3.4 and its matrix specification 3.6 . Define

Ž . � 4 � Ž . Ž . 43.26 j � i , t �J i , t � j , a set with N elements.j

Now,

ˆ ˆŽ .3.27 y �y �x �� ,� j4 � j4 � j4 � j4

where

Ž . Ž . � 43.28 y � , � n , s � j ,y� j4 n s

ˆ Ž . Ž .and similarly for x and � . Equations 3.26 and 3.27 group all of the� j4 � j4

observations on individuals employed by the same firm into the vector y , which� j4ˆ îs expressed as the deviation from the first-step estimated x� and � . The

firm-level equation is

�j

Ž . �3.29 y �F ��ˆ j� j4 � j4 � j4

�2 j

where

Ž . Ž . � 4Ž .3.30 F � , � n , s � j1 s T s �10� j4 n s 1 n s

and

ˆ ˆŽ .3.31 � �� x �� .Ž . ž /� j4 � j4 � j4 � j4 � j4

24 Since this second method is much simpler to implement than the first one, we use aspecification of the firm effect that is more complicated by including a linear spline after 10 years ofseniority.


Ž .Least squares estimation of 3.29 yields the estimator

�j�1� �Ž . Ž .�3.32 � F F F y , for j�1, . . . J .ˆ ˆj � j4 � j4 � j4 � j4

�2 j

Ž .The asymptotic distribution of the estimator in equation 3.32 is

� �j j

Ž . � �3.33 �N , � as N ��ˆj j j j� 0�2 j�2 j

where

�1 �1� � �2Ž . Ž . Ž .3.34 � �� F F � F F Fj � � j4 � j4 � j4 � j4 � j4

�1� �ˆ ˆ ˆ ˆ Ž .X var � X �var � �2 X cov � , � F F F .� j4 � j4 � j4 � j4 � j4 � j4 � j4 � j4ž /The first step of our order-dependent ‘‘firms first’’ method begins with the

ˆŽ . Ž .equations 3.16 or 3.21 defining the estimator � in the order-independentmethod. The order-dependent ‘‘firms first’’ estimator for � and � is based onconventional computational formulas applied to equation

ˆy�F��X��D��

ˆŽ .where ��F �� . The order-dependent ‘‘firms first’’ estimators are

�1� �ˆ ˆŽ . Ž .3.35 �� X M X X M y�F�Ž .D D

and

�1� �ˆ ˆ ˆŽ . Ž .3.36 �� D D D y�X��F� .Ž .Ž .The asymptotic covariance matrices of the estimators in equations 3.35 and

Ž .3.36 can be derived directly from the standard formulas and the asymptoticcovariance matrix of the order-dependent ‘‘firms first’’ estimator of � , which is

2Ž � .�just � F M F .� Z

3.4.3. Relation between the order-independent and order-dependent estimates

In the discussion of our empirical results we refer repeatedly to differentforms of the conditional estimators. In this subsection we summarize therelations among the different conditional estimators. The order-independent

Ž . Ž .estimator for � and � , equations 3.17 and 3.18 , is identical to the order-de-pendent ‘‘persons first’’ estimator for � and � . The order-dependent ‘‘persons

Ž .first’’ estimator for � is equation 3.32 . The order-independent estimator for � ,


Ž . Ž .equation 3.16 or 3.21 , is identical to the order-dependent ‘‘firms first’’estimator for � . The order-dependent ‘‘firms first’’ estimators for � and � are

Ž . Ž .equations 3.35 and 3.36 .

3.4.4. Estimation of components of the indiidual effect

Regardless of the estimator used for � , we also decompose the individualiŽeffect into a component attributable to fixed individual characteristics, u suchi

. Ž .as education , and an unobservable component, � , as shown in equation 3.2 .iTo recover the � and u � parts of the individual effect, we use the estimatedi i

îndividual effects, � , and their associated estimated sampling variances toiŽ .estimate the equation 3.2 by generalized least squares. We obtain �, whichˆ

satisfies�1�1� ˆŽ .3.37 ��N � , U diag var � U as N��ˆ ž /iž /ž /

where

u1Ž .3.38 U�

uN

ˆ 2 ˆŽ � �. � �and diag var � is a diagonal matrix � T , the asymptotic variances of �î � i iŽ .using the residual variance estimator from equation 3.24 . The estimator of � i

is

ˆŽ .3.39 � �� u �ˆ î i i

Ž Ž ..and is unbiased and asymptotic in T Chamberlain 1984 . We show below thatiˆstatistics based upon aggregating � and � to the level of the firm areî i

consistent.

3.5. Specification Checks

Ž .Because of the result in equation 3.25 , namely that the goodness of fit of themodel does not depend upon the number of auxiliary parameters used in thewithin-D or within-F step, conventional specification tests and Bayesian modelselection procedures are not applicable. Essentially, we must maintain the

Ž . Ž .conditional orthogonality assumptions 3.12 and 3.13 in order to compute anyŽ .estimates at all of equation 3.11 . Although we cannot compute a classical

Ž .specification test in the sense of Hausman 1978 , we can use those principles toderive a specification test whose distribution is known under the null hypothesis

�1� �Ž .H : �� Z Z Z F� ,0

which is the definition of the auxiliary parameter � under the conditionalŽ . Ž .orthogonality hypotheses of equations 3.12 and 3.13 .


ˆ ˆ ˆŽ .Consider the residual from equation 2.2 when � , � , and � are definedŽ . Ž .according to the order-independent estimation formulas 3.17 , 3.18 , and

Ž .3.16 , respectively,

ˆ ˆ ˆ�� y�X��D��F�˜ ž /ˆ ˆ ˆ ˆŽ . Ž .��X �� D �� Z �� M F ��Ž . Ž .Z

ˆ ˆŽ .�Z �� P F ��Ž .Z

ˆ ˆŽ .��Z �� P F ��ˆ Ž .Z

�1 �1� � � �ˆ ˆŽ . Ž .��Z �� Z Z Z F� �Z �� Z Z Z F� .ˆ Ž . Ž .Hence, under the null hypothesis

ˆ ˜Ž .��Z �� .˜ ˆˆ ˜The statistic �� is very similar to a specification test statistic since it is the

ˆdifference between the Cramer-Rao efficient estimator of �, namely �, and an˜inefficient but unbiased estimator of the same auxiliary parameter, namely �. By

ˆdirect application of the Cramer-Rao lower bound implied by the efficiency of �Ž .for the model given in equation 3.11 , we have

ˆ ˜ 2Ž .��N 0, � ��

where�1 �1� � � �2 ˆ ˆŽ . Ž . � �� Z Z Z F var � F Z Z Z �var � ,�

ˆ ˆ ˆ� � � �and var � and var � are the covariance matrices of the parameter estimators �ˆ Ž . Ž .and � , respectively, as computed in the solutions to equations 3.15 and 3.16 .

ˆ ˜The variance of �� is guaranteed to be positive semi-definite by the efficiencyôf �. Thus, a test of the specification of the model can be based upon the

distribution of �� . The statistic˜ ˆ� ��Ž . � � Ž .�� Z�Z ��˜ ˆ ˜ ˆ

2�� ,Q2��

� � � �where Q �rank Z�Z �Q. An equivalent statistic that is easier to compute is�Ž .based on the distribution of Z �� , a Q�1 random vector:˜ ˆ

� �1 �1� � ��Ž . Ž . Ž . Ž .�� Z Z Z � Z Z Z ��˜ ˆ ˜ ˆŽ .3.40 2��

� �1 �1� ��ˆ ˜ ˆ ˜Ž . Ž . Ž . Ž .�� Z Z � Z Z ��2

�� ,Q2��

� � �where Q �rank � �Q. This statistic is the only formal specification test thatwe derive for the reasonableness of the set of conditioning variables, Z.


To compare the different estimators of the � coefficients, we rely on ourŽ .consistent estimate of � and use Hausman 1978 statistics. We derive conven-

tional specification tests of the difference between the consistent estimates of �Ž .based on equation 3.9 and estimates from other methodologies, including our

conditional methods.

3.6. The Construction of Z

The role of the conditioning variables, Z, is to proxy for the covariationamong the effects represented by X, D, and F. The columns of Z should bechosen to preserve as many of the effects � as possible, recalling that each

Ž � .column of Z reduces the rank of F M F by one, while capturing as much ofZthe conditional covariance of X and D with F as possible. Since these arecompeting goals, we will rely on judgement and on the specification test in

Ž .equation 3.40 to choose a reasonable set of Z variables. We begin by notingthat every column of Z increases the computational complexity of solving the

Ž . Ž . � 2equation system 3.15 and 3.16 in proportion to N Q in terms of bothstorage and calculations. It is therefore necessary to accept some a priorirestrictions on this auxiliary design matrix Z. Second, we note that the within-Fregression in our conditional estimation procedure will not be well-defined for Zvariables that do not have within-F variation. In order to give all columns of Zsome within-F variation while, at the same time, inducing correlation with Xand D, we chose the Z variables as interactions between firm characteristicsŽ . Ž .functions of F and personal characteristics functions of X and D . Under the

Ž .specification described by equation 3.11 , none of these interactions enters themodel directly.

The columns of Z are defined as follows. Let

ÝTi xt�1 in i tx � � the within-person mean of x ,i i tTi

and

Ý fŽ i , t .� �JŽ i , t .� j4 Ž i , t . jf � � firm average of characteristic f ,j Ž i , t . jNj

where the firm characteristics are measured by taking functions of the columnsof F. In particular, firm size can be measured as a fixed constant times thenumber of person-years observed in firm j over the life of the sample, N .25 Thejindustry of firm j can be determined by applying a classification matrix A, J�K

25 We can calculate firm size in our sample using the following method. In our data, the employeesampling rate is 125th and the number of at risk years is 10; hence, the constant �2.5. Thus, inmatrix form, we convert F into a vector of firm sizes, L, as0

�� L�F e F 2.50 N 0

where e � is an N� �1 vector of ones.N


to F so that the result F A classifies each row of F , thus all N� persons-years,0 0 0into one of K unique industries. The firm characteristics actually used in ouranalysis are firm size, its square, and a 10-industry classification. The personal

Ž .characteristics actually used were labor force experience time-varying and ageŽ .at the end of schooling non-time-varying . The rows of Z were constructed as

1 sx uŽ .row i , t � � f � .i ti i JŽ i , t .

3.7. Analysis of Firm-leel Outcomes

Our analysis of firm-level outcomes requires summary statistics, by firm, ofŽ .the effects estimated from equation 2.2 . Although we use several different

estimators for these effects, we always use the same aggregation formulas; so,we have shown those formulas using generic estimators for the underlyingparameters.

First consider firm-level averages of the person effects � and � ,i i

1ˆ ˆŽ .3.41 � � �Ýj iNj Ž . � Ž . 4i , t � J i , t �j

and

1� � � .ˆ ˆÝj iNj Ž . � Ž . 4i , t � J i , t �j

We use the asymptotic distribution for � :ˆj

Ž . 23.42 � �N � , � , as N ��ˆ ž /j j � jj

where

N 2j �11 � T �1� i � �2 ˆ� � 1� u U diag var � U uÝ ž /� i i iž /2 2j TN �ij �i�1

ˆ 26Ž .and for � not shown . Similarly, the firm-level average education effect isjgiven by

1Ž .3.43 u �� u �ˆ ˆÝj iNj Ž . � Ž .i , t � J i , t �j

Ž .with asymptotic distribution based upon 3.37 . In all our asymptotic results wehold constant the distribution of firm sizes. Thus as N, N ��, we assume thatjtheir ratio N N goes to a nonzero constant.j

26 ˆThe formula for the symptotic distribution of � is identical to the one for � with theˆj jquadratic form in u removed.i


We consider next the statistical relation between firm-level outcomes and ourmeasures of firm-level compensation policy. Our basic model is

�� u � � � � qŽ .3.44 p � ��j j j j 2 j jj j�

� �where j�1, . . . J, p is any firm-level outcome, � u � � � � is a vector ofj j j j j 2 j

firm-level compensation measures, � is a vector of parameters of interest, q is ajvector of other firm-level variables, � is a vector of parameters associated withq , and � is a zero-mean homoscedastic statistical error.27 In the regressionj j

analysis, firm-level outcomes and firm-level compensation variables were mea-sured using data from two independently drawn samples. However, the firm-levelcompensation variables derived from our individual sample are estimated re-

ˆgressors. Consequently, we must allow for the estimation errors in � , u �, � ,ˆ ˆj j j� , and � in our assessment of the precision of the estimation of firm-levelˆ ˆj 2 j

28 Ž .equations. Equation 3.44 becomes

�ˆŽ . � u � � � � q3.45 p � ˆ ˆ ˆ ˆj j j j 2 j jj �

� u � � � �� j j j j 2 jžˆ� u � � � �� ˆ ˆ ˆ ˆj j j j 2 j j/

ˆŽ� � � �.where � u � � � � � � u � � � � � is the error associated withˆ ˆ ˆ ˆj j j j 2 j j j j j 2 j

the first-step estimation of the firm-level compensation measures.29 In order toŽ .derive the error covariance matrix for equation 3.45 , let

� ˆˆ � u � � � � qP � ˆ ˆ ˆ ˆž / j j j j 2 j jj j

and

� ˆˆ � u � � � � � .ˆ ˆ ˆ ˆj j j j 2 jj

27 This is the most general specification, corresponding to the parameterization of the firm effectŽ .m�3 used in our order-dependent ‘‘persons first’’ method. In some of our firm-level analyses theterms involving � do not appear because the underlying firm effects were of lower dimension2 jŽ .m�2 .

28 ˆThe firm-level regressor x � also contains some measurement error, in principle; however, thejˆvector � is estimated with such precision that we do not carry along its estimated covariance matrix

ˆŽ .including its estimated covariance matrix with � , u �, � , � , and � in these calculations. Hence,ˆ ˆ ˆ ˆj j j j 2 jˆwe place x � in the list of q .j j

29 Ž .We adopt the model of Pagan 1984 ; namely, that the regression of interest relates a functionof the individual-level data and several firm-level parameters to the other measured firm-level

ˆŽ� � � �.outcomes. We account for the estimation error � u � � � � � � u � � � � explicitly,ˆ ˆ ˆ ˆj j j j 2 j j j j j 2 jbut we do not add an additional measurement error. Thus, for example, we assert that the outcomep depends upon � and not upon � �� , where � is an independent measurement error.j j j j j


Ž .Now, equation 3.45 can be re-expressed in a first order approximation around asj

��Ž . Ž .3.46 p �P ��j j j j�

where� Ž .� P � j j �ˆ� � � �� .ž /j j j j��j

Ž .The variance of the regression error term for equation 3.46 consists of theˆcomponent due to the estimation error in P plus the component due to � :j j

�� P � Pj j �� ˆŽ . � � � �� 3.47 var � � var �var ��j j j�� j j

ˆ� �where the components of var are defined in the derivations above. WejŽ .estimate equation 3.46 using generalized least squares based upon the errorŽ .variance in equation 3.47 .

4. DATA DESCRIPTION

In this section we describe the important institutions of the French labormarket and compare some simple statistical models of wage determination inFrance and the United States. The wage regressions demonstrate that, eventhough French and American labor market institutions are quite different, thereare strong similarities in the way compensation is related to labor marketobservables in the two countries. Next, we lay out the sample design of ourFrench data and describe the process we used to create an analysis sample.Finally, we present all of the variable definitions. Summary statistics appear inthe Data Appendix.

4.1. The French Labor Market

Ž .During the sample period from the mid-seventies to the end of the eighties ,the French labor market was characterized by stable employment, whereas overthis period employment increased by 25% in the United States. GDP growth inboth countries was more or less identical, implying faster productivity growth inFrance. In addition, the employment-population ratio in France shrank while itwas growing in the U.S.; as a reference, employment-population ratios were thesame in France and the United States in the mid-sixties. In particular, the

Ž .employment-population ratio fell dramatically for young workers below 25 asŽ . 30well as for older workers above 55 . The prevailing view�challenged in Card,

30 Ž .See Card, Lemieux, and Kramarz 1996 , for a more detailed analysis of French labor marketoutcomes in comparison with those of the United States and Canada.


Ž .Kramarz, and Lemieux 1996 �is that wage rigidities, examples of which arepresented in the following paragraphs, have destroyed jobs in France. Neverthe-less, even though wage-setting institutions differ, wage setting outcomes in thetwo labor markets share many features.

French employment in the 1970s was characterized by centralized collectiveŽ .bargaining convention collective de branche , in which different industrial

sectors had collective agreements that were negotiated by groups of unions andemployers associations, and these agreements were binding on the negotiatingparties. The complete agreement was then typically extended to cover the entire

Ž .industry or region by the Ministry of Labor and was thereby made binding onŽworkers and firms that were not party to the original negotiation see Margolis

Ž ..1993 . More than 95% of the work force was covered by these collectivebargaining agreements at the end of the 1980s, while union membership wasapproximately 10%. The collective agreements specified a set of minimum wagesand wage progressions for the occupational categories covered by the negotia-

Ž . Žtions sometimes called a wage grid . Beginning in 1982, the ‘‘lois Auroux’’ a setof revisions to the body of labor law named after the Minister of Labor at the

.time required firms with at least 50 employees to negotiate firm-level collectiveŽ .agreements accords d’entreprise . Although firms were explicitly not obligated

to actually conclude an agreement, the percentage of the work force covered byŽfirm-level agreements grew to over 30% by the mid-1980s see Abowd and

Ž . Ž ..Kramarz 1993 and Cahuc and Kramarz 1997 . The law imposed that thefirm-level agreements could only improve the conditions stated in the industrialagreement, a result being that, over time, the firm-level agreements havebecome more relevant for wage determination than the industry agreements.

Since 1951, French industry has also been subject to a national minimumŽ .wage called the SMIC since the revisions to the relevant law in 1971 that is

indexed to the rate of change in consumer prices and to the average blue-collarwage rate. Although more than 90% of French workers are covered by industrial

Ž .agreements throughout our analysis period 1976�1987 , the regular increases inŽthe national minimum wage in particular those driven by the indexation to the.average blue-collar wage rate outpaced contract renegotiations, and the lowest

rungs on the pay scales in most industry contracts for most occupations endedup below the national minimum in 1985. When this occurs, it is the nationalminimum wage, and not the collectively bargained wage, that binds.

Even though the French institutional arrangements seem to differ widely fromthose prevailing in the United States, wage-setting outcomes in the two coun-tries share many features. For instance, manufacturing operative wages, when

Žmeasured in purchasing power parity, are not very different see Abowd andŽ ..Bognanno 1995 . However, the ratio of the minimum wage to the average wage

Žfell sharply in the U.S. while it rose modestly in France during the eighties seeŽ ..Card, Kramarz, and Lemieux 1996 . Roughly 7% of French employed young

Ž .workers 30 years old and under , and 6% of American employed young workersŽare paid at the minimum during the same period see Abowd, Kramarz,

Ž ..Lemieux, and Margolis forthcoming . Even though total labor costs at theminimum wage are higher in France than in the U.S. due to employee- and


employer-paid payroll taxes and other nonwage compensation costs, a 1%increase in the minimum wage induces roughly a 2% decrease in employment of

Žyoung people in both countries Abowd, Kramarz, Lemieux, and MargolisŽ ..forthcoming .

To further assess potential differences in wage setting, we ran two simpleŽwage regressions using comparable household surveys the Enquete Emploi forˆ. 31France and the Current Population Survey for the U.S. . Table II presents our

estimation results. Our models show that the same set of regressors has more orless the same explanatory power for wages in both the French and American

Ždata roughly 37% for men in both countries, 32% for women in France and.24% in the U.S. . Returns to one additional year of education are 6.1% for men

and 7.2% for women in the U.S. while they are 7.7% for men and 8.8% forwomen in France, with the difference between the sexes being identical. Returnsto experience differ slightly, with the curvature of the quartic in experienceimplying a more hump-shaped profile in the U.S. Finally, the gender wage gap inthe initial year is roughly equal in both countries, although it decreases over thesample period in the U.S. and is basically stable in France during the eighties.

Other examples of such similarities in wage-setting outcomes abound. Card,Ž .Kramarz, and Lemieux 1996 have shown that the fraction of workers using

computers is roughly the same in the two countries. Furthermore, returns tonew technologies, and in particular computer use, are identical in the two

Ž . Ž .countries. Estimates in Krueger 1993 , in Entorf and Kramarz 1997 , or Entorf,Ž .Gollac, and Kramarz forthcoming show that computer users are better com-

Ž .pensated than nonusers by the same amount 15% . Krueger and SummersŽ .1987 also show that inter-industry wage differentials in France are highlycorrelated with American inter-industry wage differentials.

4.2. Description of the DAS

Ž .Our main data source is the ‘‘Declarations Annuelles des Salaires’’ DAS , a´large-scale administrative database of matched employer-employee information

Žcollected by INSEE Institut National de la Statistique et des Etudes.Economiques and maintained in the Division des Revenus. The data are based

upon mandatory employer reports of the gross earnings of each employeesubject to French payroll taxes. These taxes apply to all ‘‘declared’’ employeesand to all self-employed persons, essentially all employed persons in the econ-omy.

The Division des Revenus prepares an extract of the DAS for scientificanalysis, covering all individuals employed in French enterprises who were bornin October of even-numbered years, with civil servants excluded.32 Our extract

31 Similar results are also found using cross-sections of matched worker-firm data for the twoŽ Ž ..countries see Abowd, Kramarz, Margolis, and Troske 1998 .

32 Ž .Meron 1988 shows that individuals employed in the civil service move almost exclusively toother positions within the civil service. Thus, the exclusion of civil servants should not affect ourestimation of a worker’s market wage equation. Employees of the state-owned firms are present inour sample, however.


TABLE II

COMPARISON OF LEAST SQUARES ESTIMATES OF WAGE DETERMINATION IN FRANCE

AND THE UNITED STATES 1982�1987

France United States

Men Women Men Women

Variable Mean OLS Results Mean OLS Results Mean OLS Results Mean OLS Results

Intercept 1.000 1.365 1.000 1.163 1.000 0.534 1.000 0.380Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .0.000 6.746E-3 0.000 8.190E-3 0.000 5.614E-3 0.000 5.679E-3

Years of 10.726 0.077 11.325 0.088 11.880 0.061 12.300 0.072Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .Education 3.659 2.848E-4 3.267 3.998E-4 2.391 3.521E-4 2.149 3.712E-4

Experience 20.722 0.058 19.048 0.060 15.894 0.112 16.036 0.082Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .12.222 1.228E-3 12.163 1.435E-3 12.311 9.219E-4 12.323 8.899E-4

2Experience 5.788 �0.104 5.108 �0.188 4.042 �0.506 4.090 �0.436Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .100 5.875 9.095E-3 5.533 1.133E-2 5.251 8.487E-3 5.077 8.482E-3

3Experience 18.915 �0.007 16.207 0.018 12.611 0.102 12.544 0.093Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .1,000 26.344 2.554E-3 23.730 3.355E-3 21.849 2.811E-3 20.514 2.898E-3

Experience4 68.009 0.002 56.700 0.001 43.914 �0.007 42.506 �0.007Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .10,000 120.682 2.397E-4 103.687 3.316E-4 92.920 3.037E-4 84.740 3.232E-4

1982 0.175 0.036 0.167 0.027 0.163 0.072 0.160 0.019Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .0.380 3.001E-3 0.373 3.784E-3 0.370 2.715E-3 36.707 2.596E-3

1983 0.170 0.018 0.166 0.006 0.162 0.049 0.160 0.015Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .0.375 3.020E-3 0.373 3.783E-3 0.369 2.707E-3 0.367 2.579E-3

1984 0.166 0.019 0.164 0.012 0.164 0.032 0.162 0.000Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .0.372 3.033E-3 0.371 3.793E-3 0.370 2.679E-3 0.368 2.557E-3

1985 0.165 0.005 0.165 �0.001 0.167 0.018 0.166 �0.002Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .0.371 3.040E-3 0.371 3.785E-3 0.373 2.658E-3 0.372 2.534E-3

1986 0.162 0.023 0.168 0.018 0.174 0.015 0.175 0.004Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .0.369 3.051E-3 0.374 3.767E-3 0.379 2.601E-3 0.380 2.479E-3

Paris Region 0.210 0.168 0.240 0.158 � � � �Ž . Ž . Ž . Ž .0.407 2.147E-3 0.427 2.567E-3 � � � �

Northeast U.S. � � � � 0.210 �0.046 0.217 �0.057Ž . Ž . Ž . Ž .� � � � 0.408 2.496E-3 0.412 2.393E-3

Midwest U.S. � � � � 0.263 �0.039 0.273 �0.088Ž . Ž . Ž . Ž .� � � � 0.440 2.309E-3 0.446 2.222E-3

Southern U.S. � � � � 0.296 �0.143 0.289 �0.128Ž . Ž . Ž . Ž .� � � � 0.457 2.206E-3 0.453 2.151E-3

Observations 165,036 165,036 126,320 126,320 259,297 259,297 259,266 259,2662Adjusted R 0.3866 0.3254 0.3626 0.2428

Ž . Ž .Sources: Enquete Emploi 1982�1987 for France and NBER outgoing rotation group CPS extracts 1982�1987 for theÛnited States. Notes: Standard DeviationsErrors in Parentheses. Both regressions included only individuals between 16

Ž . Žand 60 years old, inclusive. Both regressions used the sample weights. Experience is measured as age � age at the end of. Ž . Ž .schooling in France and age � years of schooling � 5 in the United States.

runs from 1976 through 1987, with 1981 and 1983 excluded because theunderlying administrative data were not sampled in those years. The initial dataset contained 7,416,422 observations. Each observation corresponded to a uniqueindividual-year-establishment combination. An observation in this initial DAS

Ž .file includes an identifier that corresponds to the employee called ID below , anŽ .identifier that corresponds to the establishment SIRET and an identifier that

Ž .corresponds to the parent enterprise of the establishment SIREN . For eachobservation, we have information on the number of days during the calendaryear the individual worked in the establishment and the full-timepart-time


status of the employee. For each observation corresponding to an individual-year-establishment, in addition to the variables listed above, we have informa-tion on the individual’s sex, date and place of birth, occupation, total netnominal earnings during the year and annualized gross nominal earnings duringthe year for the individual, as well as the location and industry of the employingestablishment.

4.3. Obseration Selection, Variable Creation and Missing Data Imputation

4.3.1. Aggregation across establishments

The creation of the analysis data set involved the selection of desiredindividuals, the aggregation of establishment-level data to the enterprise level,and the construction of the variables of interest from the variables already in

Žthe data set. We selected only full-time employees sample reduced to 5,966,620.observations . We then created a single observation for each ID-year-SIREN

combination by aggregating within ID and year over SIRETs in the sameSIREN. For each ID-year-SIREN, we summed total net nominal earnings andtotal days worked over all SIRETs. We assigned to the observation the occupa-tion, location, and industry that corresponded to the establishment in which theindividual worked the largest number of days during the year. This reduced thenumber of observations to 5,965,256. We then selected the enterprise at which

Žthe individual had worked the largest number of days during that year sample.reduced to 5,497,287 observations . The aggregation of total number of days

worked across all establishments occasionally yielded observations for which theŽ .total number of days worked was greater than 360 the maximum permitted . In

these cases, we truncated days worked at 360. We then calculated an annualizednet nominal earnings for the ID-year-SIREN combination. We eliminated allyears of data for individuals who were younger than 15 years old or older than

Ž65 years old at the date of their first appearance in the data set sample reduced.to 5,325,413 observations .

4.3.2. Total compensation costs

The dependent variable in our wage rate analysis is the logarithm of realannualized total compensation cost for the employee. To convert the annualizednet nominal earnings to total compensation costs, we used the tax rules and

Žcomputer programs provided by the Division des Revenus at INSEE Lheritier,´.internal, undated INSEE communication to compute both the employee and

Žemployer share of all mandatory payroll taxes cotisations et charges salariales:.employe et employeur . Annualized total compensation cost is defined as the´

sum of annualized net nominal earnings, annualized employee payroll taxes, andannualized employer payroll taxes. Nominal values were deflated by the con-sumer price index to get real annualized net earnings, and real annualized totalcompensation cost. We eliminated 61 observations with zero values for annual-

Ž .ized total compensation cost remaining sample 5,325,352 .


4.3.3. Education and school-leaing age

Our initial DAS file did not contain education information. We used supple-Žmentary information from the permanent demographic sample Echantillon

.Demographique Permanent, EDP available for 10% of the DAS, to impute the´level of education for all remaining individuals in the DAS. The EDP includesinformation on the highest degree obtained. There were 38 possible responses,including ‘‘no known degree.’’ These responses were grouped into eight degree-level categories as shown in Data Appendix Table B1. Using these eightcategories as the dependent variable and data available in the DAS, we ranseparate ordered logits for men and women. We used the data corresponding tothe earliest date that an individual appeared in our sample to estimate thesemodels. We used the same data and the estimated coefficients from theseordered logit models to impute the probability of obtaining each of the eightdifferent aggregated degrees for the individuals in the DAS who were not partof the EDP. We used the actual value of the eight degree aggregates for theEDP sample members. Thus, a random 10% sample of the DAS individualshave true education and the remaining 90% have the probability of obtainingeach of the eight degree aggregates. EDP sample statistics for the men are inData Appendix Table B2, and those for the women are in Table B3. Theestimated logit equations are in Table B4 for men and Table B5 for women.33

To calculate school-leaving age we used Table 14 in CEREQ-DEP-INSEEŽ .1990 , which provides the average age of termination for each French diplomaseparately for men and women in 1986. Using the probability of each degree

Žcategory and the average school-leaving age for degrees in that category the.ages were fairly homogeneous within categories , we calculated expected

school-leaving age.

4.3.4. Total labor market experience

For the first year in which individuals appear in the sample, we calculatepotential labor market experience as age at the beginning of the year less ourestimate of school-leaving age. In all subsequent years, total labor marketexperience is accumulated using the individual’s realized labor force history.Our algorithm was the same for both labor force experience and seniority. It

33 Ž .We considered, and rejected, the possibility of using a Rubin 1987 style multiple imputationŽ .algorithm for the missing schooling variable for the following reasons: 1 since schooling does not

time-vary, and since our conclusions are completely unaffected by whether we remove a schoolingŽ .effect from the person effect or not � as compared with � , we did not want to bear the

Ž .computational burden associated with these methods for such a small return; 2 the schoolingvariable is substituting for occupational category, a more common control variable in Frenchearnings equations because of the educational qualifications that define the occupational categories,in our models in order to make the analysis more comparable to the vast American literature that

Ž .uses schooling rather than occupation and defines person effects with a schooling effect removed; 3conditioning the imputation on the observed value of the compensation variable, as these methodsrequire, would focus attention on the imputation procedure and detract from our main focus.


accounts for the holes caused by the fact that the administrative data were notavailable for 1981 and 1983. See the Data Appendix for details.

4.3.5. Job seniority

Individuals fell into two categories with respect to the calculation of jobŽ .seniority employer-specific experience : those for whom the first year of obser-

vation was 1976 and those who first appeared after 1976. For those individualswhose first observation was in 1976, we estimated the expected length of thein-progress employment spell by regression analysis using a supplementary

Žsurvey, the 1978 Salary Structure Survey Enquete sur la Structure des Salaires,ˆ.ESS . In this survey, respondent establishments provided information on senior-Ž .ity in 1978 , occupation, date of birth, industry, and work location for a

scientific sample of their employees. Using the ESS information, we estimatedseparate regressions for men and women to predict seniority for in-progressspells in 1976. The coefficients from these regressions were used to calculateexpected job seniority in 1976 for DAS individuals whose first observation was in1976. The dependent variable in the supplementary ESS regressions was currentseniority with the employer and the explanatory variables were date of birth,

Ž . Ž .occupation 1-digit , region of employment metropolitan Paris , and industryŽ .NAP 100, approximately 2-digit . Results of the seniority regressions are shown

Ž . Ž .in equations 8.1 for men and 8.2 for women in the Data Appendix. We usedthe results of these seniority regressions to impute levels of job seniority in 1976for the left-censored DAS individuals first observed in 1976. Details are pro-vided in the Data Appendix.

4.3.6. Elimination of outliers

After calculating all of the individual-level variables, we eliminated observa-tions for which the log of the annualized real total compensation cost was morethan five standard deviations away from its predicted value based on a linearregression model with independent variables: sex, region of France, experience

Ž Ž . .and education see equation 8.4 in the Data Appendix . This gives us theanalysis sample of 5,305,108 observations.

Table B7 in the Data Appendix shows the basic summary statistics, by sex, forthe individual-level data. The usable sample consists of 3,434,530 observationson 711,518 men and 1,870,578 usable observations on 454,787 women. The basicindividual-level variables are: sex, labor force experience, region of France,education, and seniority. Note that about 30% of the sample has no knowneducational attainment. For 74% of the individuals, there are enough observa-tions in the sample to permit estimation of a distinct firm-effect.34

34 The individuals from firms with fewer than 10 observations in the sample were pooled and asingle firm-level regression was used to estimate their firm effects.


4.4. Construction of the Firm-Leel Data

4.4.1. DAS-based firm-leel aerages

For our firm-level analyses we calculated the aggregates � , u � and theirˆ ˆj jrespective sampling variances based on the � and u � estimated according toˆ î ithe conditional estimation methods laid out in Section 3 above. The estimated

ˆparameters � , � , and � have unique values for a given enterprise, byˆ ˆj j 2 j

construction. In cases where any one of the following three conditions failed:ˆ ˆ�3�� 3 or �2�� 2 or �2�� 2, we set � , � , and � equalˆ ˆ ˆ ˆ ˆj j j 2 j j j 2 j

to the values estimated in the pooled model for the firm effects for all firms with10 or fewer observations.

4.4.2. Other firm-leel data sources

Ž .The primary source of our firm-level data is the INSEE 1989, 1990a�cŽ .enterprise sample Echantillon d’Entreprises, EE , a probability sample of

Ž .French firms synonymous with enterprises for our purposes . The EE data setprovides the sampling frame for the firm-based part of this paper. The universefor the sample is the annual report on profitability and employment by enter-

Ž .prises Benefices Industriels et Commerciaux, BIC and the annual survey of´ ´Ž .enterprises Enquete Annuelle d’Entreprises, EAE . To construct the EE, firmsˆ

with more than 500 employees were sampled from the BIC with probability 1;firms with 50 to 499 employees were sampled from the BIC with probabilitiesranging from 14 to 12 depending upon the industry, and smaller firms weresampled from the BIC with probability 130. All firms responding to the BICwere at risk to be sampled exactly once. Hence, the EE is dynamically represen-tative of French enterprises in all sectors except the public administration

Ž .sector. We use the sampling weight non-time-varying and the variables de-scribed below, averaged over the period 1978 to 1988 for all available years,from the EE.

4.4.3. Firm-leel employment and capital stock

The measure of employment, in thousands of workers, is full-time employ-Ž .ment in an enterprise as of December 31 prior to 1984 and the annual average

Ž .full-time employment 1984 and later as found in the BIC. We took the meanof this variable over all years that the firm appeared in the sample.

Ž .Total capital in the enterprise is defined as the sum of debt dettes andŽ .owners’ equity fonds propres d’entreprise . Our capital measure is equal to

Ž .total assets actif total in French accounting systems. The information wastaken directly from the BIC for every firm-year. We deflated the capital stockusing an industry-specific, annual index of the price of capital from the INSEE

Žmacroeconomic time series data Banque de Donnees Macroeconomiques,´ ´.BDM . Our measure of real total capital is defined as total assets divided by the

Ž .industry-specific price index of physical capital in millions of 1980 FF , averaged


over all available years for the firm. The capital labor ratio is defined as realtotal capital divided by total full-time employment. We also averaged thisvariable over all available years for the firm.

4.4.4. Real operating income per unit of capital

ŽWe used the BIC to obtain the operating income excedent brut d’exploita-´.tion, EBE , for each firm in each year that it appeared in the firm sample. The

Ž .formula used to calculate the EBE is shown in equation 8.6 in the DataŽAppendix. The EBE was deflated by the value added price index prix de valeur

. Žajoutee also found in the BDM, to yield real operating income in millions of´.1980 FF . Real operating income was divided by real total capital to yield real

operating income per unit of capital, stated as a proportion. We also took themean of this variable over all available years for the firm.

4.4.5. Real alue added inclusie of labor costs

ŽTo calculate the real value added inclusive of labor costs valeur ajoutee´.reelle brute au cout des facteurs , we divided the employer’s compensation costs´ ˆ

Ž . Ž .frais de personnel in the BIC thousands of FF by the consumer price indexŽ .indice des prix a la consommation from the BDM to yield the employer’s real`

Ž .compensation cost in millions of 1980 FF . The result was added to realoperating income, as defined above in Section 4.4.4, to yield the real value added

Ž .inclusive of labor costs in millions of 1980 FF . Real value added inclusive oflabor costs was divided by total employment to yield real value added inclusive

Ž .of labor costs per worker in thousands of 1980 FF . We then took the mean ofthis variable over all of the years that the firm appeared in the EE sample.

4.4.6. Employment structure

ŽThe variables concerning the occupational structure of employment propor-tion of engineers, technicians and managers in the work force and proportion of

. Žskilled workers were created from the employment structure survey Enqueteˆ.sur la Structure des Emplois, ESE , which is an annual administrative data base

Ž .of the detailed 4-digit occupational structure of all establishments with morethan 20 employees. The occupational structure of the firm, measured in theESE, was merged with the EE using the firm identifier and the survey year.Engineers, technicians and managers were coded using the simplified occupation

Ž .classifications 1-digit equivalents for individuals in categories 30 and 40. Theproportion of skilled workers in the work force was calculated from the ESEusing the simplified occupation classification for individuals in categories 50 and61. Both variables were expressed as a ratio to total employment and averagedover all the available firm-years. The omitted category is unskilled workers,which would include all other codes.


5. ESTIMATION RESULTS

5.1. Oeriew

We present our statistical results in three main parts. Tables III�VI presentdetailed results from the analysis of the matched employer-employee microdata.Table III shows the regression coefficients for men and women from all of theestimation methods described above as well as results from standard specifica-

Ž .tions based upon incomplete parameterizations of equation 2.2 . Table IVpresents summary statistics for men and women for all of the components ofcompensation in our complete model and for the two estimation methods uponwhich we focus most of our subsequent attention. Table V is a diagnostic tableof the correlations among the same components of compensation when we varythe method of estimation. Table VI is a table of correlations among all thecomponents of compensation for our two chosen estimation methods. TablesVII and VIII present our statistical analysis of the inter-industry wage differen-tial and the firm-size wage effect, respectively. Tables IX�XII present theresults of analyses conducted at the firm level. Table IX shows summarystatistics for the firm-level variables, including those we created from thematched microdata. Table X presents the results of our analysis of firm-levelprofitability and productivity. Table XI presents our analysis of firm-level factorsof production and compensation components. Table XII presents the results ofa survival analysis using the firm-level data.

5.2. Results from the Estimation of the General Compensation Equation

5.2.1. Specification of the compensation equation

Table III presents a summary of the estimates of � , the coefficients on thetime-varying individual characteristics, for our consistent estimation method, ourconditional estimation method with persons first, and ordinary least squaresunder a variety of different assumptions about the included person and firmeffects, separately for men and women. The results labelled ‘‘Consistent Method

Ž .Person & Firm Effects’’ were calculated according to equation 3.9 . The resultslabelled ‘‘Conditional Method Persons First’’ were calculated using the formula

Ž .found in equation 3.17 . The � coefficients obtained by the order-dependentmethod with persons first, conditional on Z, and those obtained by the order-in-dependent method, conditional on Z, are mathematically identical. The columnlabelled ‘‘Least Squares No PersonFirm Effects’’ presents the estimates ob-tained when we set all person effects, D� , firm effects, F� , and conditioningeffects, Z�, jointly to zero. Next, in the column labelled ‘‘Within Persons NoFirm Effects,’’ we present results obtained when we retain the person effects,D� , but set all firm effects, F� , and conditioning effects, Z�, jointly to zero. Inthe column labelled ‘‘Within Persons Limited Firm Effects,’’ we present �coefficients estimated when we retain all person effects, D� , choose a set ofeffects Z equal to the columns of F corresponding to the 115 largest employers


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TABLE IV

DESCRIPTIVE STATISTICS FOR COMPONENTS OF LOG REAL TOTAL ANNUAL

COMPENSATION BY SEX FOR 1976 TO 1987

Men WomenVariable Definition Mean Std. Dev. Mean Std. Dev.

Ž .Log Real Annual Compensation, 1980 FF 4.3442 0.5187 4.0984 0.4801

Order-Independent Method

x� , Predicted Effect of x Variables 0.3890 0.1489 0.2849 0.1144� , Individual Effect Including Education 3.9552 0.4475 3.8135 0.3930

Ž .� , Individual Effect Unobserved Factors 0.0000 0.4051 0.0000 0.3771u�, Individual Effect of Education and Sex 3.9552 0.1902 3.6893 0.1107

Ž .� , Firm Effect Intercept and Slope �0.0363 0.4642 0.0665 0.5116�, Firm Effect Intercept �0.1367 0.4532 �0.0235 0.4967� , Firm Effect Slope 0.0149 0.0503 0.0172 0.0531

Order-Dependent Method: Persons First

x� , Predicted Effect of x Variables 0.4261 0.1383 0.3234 0.1120� , Individual Effect Including Education 3.9160 0.4387 3.7776 0.3843

Ž .� , Individual Effect Unobserved Factors 0.0000 0.3947 0.0000 0.3639u�, Individual Effect of Education and Sex 3.9160 0.1915 3.7776 0.1238

Ž .� , Firm Effect Intercept and Slope 0.0028 0.0685 �0.0039 0.0566�, Firm Effect Intercept 0.0031 0.1044 �0.0072 0.0969� , Firm Effect Slope �3.37e-05 0.0335 8.28e-04 0.0326� , Firm Effect Slope Change at 10 years �5.36e-04 0.0542 �1.64e-03 0.05742

Other Estimation Methods

Ž . Ž .Seniority coefficient, Least Squares 0.0118 0.0001 0.0141 0.0001Ž .Standard Error

Ž . Ž .Seniority coefficient, within Persons 0.0033 0.0001 0.0024 0.0001Ž .Standard Error

Ž . Ž .Seniority coefficient, within Firms 0.0078 0.0001 0.0102 0.0001Ž .Standard Error

Ž . Ž .Seniority coefficient, within Industry 0.0090 0.0001 0.0121 0.0001Ž .Standard Error

Ž . Ž .Seniority coefficient, within Size Class 0.0097 0.0001 0.0126 0.0001Ž .Standard Error

� , Firm Effect Slope, 115 largest firms 0.0013 0.0065 0.0014 0.0076� , Firm Effect Slope, Consistent estimates 0.0116 0.0342 0.0138 0.0352

Notes: Seniority coefficients with standard errors were estimated in the same models reported in Table III. All otherstatistics are the means and standard deviations based upon the sample of 5,305,108 observations except for the FirmEffect Slope in the 115 largest firms, which are statistics based on 695,077 observations.

Ž .in our data firm-specific intercepts and seniority slopes , and set all remainingfirm effects, F� , to a single common effect. Thus, this column shows estimatesof a model in which 695 thousand of our 5.3 million observations have separate,

Ž .firm effects firm-specific intercepts and seniority slopes and all remainingobservations are pooled into a single artificial ‘‘firm,’’ which had its own


TABLE V

CORRELATIONS AMONG THE COMPONENTS OF PERSON AND FIRM HETEROGENEITY AS

ESTIMATED BY THE ORDER-INDEPENDENT, ORDER-DEPENDENT, FULL LEAST SQUARES

ON THE 115 LARGEST FIRMS, AND CONSISTENT METHODS

Simple Correlation with:

Full Least SquaresSource of Estimate of Parameter Order-Independent Estimates on the Consistentthe Indicated Effect Name Estimates Order-Dependent Estimates 115 Largest Firms Estimates

Persons FirstFirm Effects � � � � � � � �2Order-Independent � 1.0000 �0.0718 0.1553 �0.0837 0.0188 0.0888 0.2800 0.0361

Estimates � �0.0718 1.0000 �0.2202 0.5300 �0.0077 �0.3276 0.3126 0.0907Order-Dependent � 0.1553 �0.2202 1.0000 �0.5625 0.2562 0.6659 �0.0231 �0.1810

Estimates � �0.0837 0.5300 �0.5625 1.0000 �0.2094 �0.6580 0.2739 0.1358Ž .Persons First � 0.0188 �0.0077 0.2562 �0.2094 1.0000 0.5492 0.0293 �0.01262

Full Least Squares � 0.0888 �0.3276 0.6659 �0.6580 0.5492 1.0000 �0.1841 �0.1964Estimates Using the � 0.2800 0.3126 �0.0231 0.2739 0.0293 �0.1841 1.0000 0.5106115 Largest Firms

Consistent Estimates � 0.0361 0.0907 �0.1810 0.1358 �0.0126 �0.1964 0.5106 1.0000

Firms First

Person Effects � � �

Order-Independent � 1.0000 0.5833 0.9896Estimates

Order-Dependent � 0.5833 1.0000 0.5983EstimatesŽ .Firms First

Full Least Squares � 0.9896 0.5983 1.0000Estimates Using the115 Largest Firms

Notes: N� 5,305,108, except for Full Least Squares Estimates Using the 115 Largest Firms where N � 695,077.Source: Authors’ calculations based on the DAS.

intercept and seniority slope. Table III also shows, in the column labelled‘‘Within Firms No Person Effects,’’ the results obtained when we estimate a

Ž .model where we retain all firm effects, F� intercepts only , and set all personeffects, D� , and all conditioning effects, Z�, to zero. The last two columnspresent results obtained from estimating a model with person effects and firmeffects jointly set to zero, and with the functions Z of the form Z�FA, where

ŽA generates 84 industry effects and is as defined in Section 2.1 ‘‘Within.Industry No Person Effects’’ , and of the form Z�FS, with the matrix S

generating 25 firm-size classes based on the firm sizes constructed as describedŽ .in Section 3.6 ‘‘Within Firm Size No Person Effects’’ .

ŽAll of the estimation methods that include person effects consistent method,conditional method with persons first or order independent, within persons

.without firm effects, and within persons with limited firm effects are able toexplain a similar fraction of the variance�between 77% and 83%. In contrast,

Žall of the results that exclude person effects ordinary least squares, within firms,.within industries, and within firm-size categories give results similar to the


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TABLE VII

GENERALIZED LEAST SQUARES ESTIMATION BETWEEN INDUSTRY WAGE EFFECTS AND

INDUSTRY AVERAGES OF FIRM-SPECIFIC COMPENSATION POLICIES

Standard Standard StandardIndependent Variable Coefficient Error Coefficient Error Coefficient Error

Based on Order-Independent Estimates

Ž . Ž .Industry Average � 1.0390 0.0023 1.0053 0.0022Ž . Ž .Industry Average � �0.0220 0.0006 0.0683 0.0005Ž . Ž . Ž .Intercept 3.3023 0.0019 3.3031 0.0019 3.0935 0.0018

2R 0.8487 0.8425 0.0682

Based on Order-Dependent Estimates: Persons First

Ž . Ž .Industry Average � 0.8011 0.0019 0.8324 0.0017Ž . Ž .Industry Average � 0.2410 0.0151 �0.6659 0.0150Ž . Ž . Ž .Intercept 3.1126 0.0019 3.1088 0.0018 3.0687 0.0019

2R 0.9580 0.9213 0.2486

Notes: The dependent variable is the 84 industry-effects estimated by least squares controlling for labor force experienceŽ . Ž . Ž .through quartic , seniority, region, year, education eight categories and sex fully interacted . See Table III for theregression results. The independent variables are the industry averages for the indicated firm-specific compensation policy,adjusted for the same independent variables. The time period is 1976�1987.

TABLE VIII

GENERALIZED LEAST SQUARES ESTIMATES OF THE RELATION BETWEEN FIRM-SIZE WAGE EFFECTS

AND FIRM-SIZE CATEGORY AVERAGES OF FIRM-SPECIFIC COMPENSATION POLICIES



Ž . Ž .Firm-Size Category Average � 1.2222 0.0043 1.3245 0.0041Ž . Ž .Firm-Size Category Average � 0.2233 0.0026 0.4278 0.0025Ž . Ž . Ž .Intercept 3.7397 0.0022 3.6737 0.0021 3.5215 0.0021

2R 0.9604 0.8960 0.2559


Ž . Ž .Firm-Size Category Average � 1.1372 0.0045 1.3224 0.0042Ž . Ž .Firm-Size Category Average � 0.9217 0.0085 1.7395 0.0079Ž . Ž . Ž .Intercept 3.6370 0.0022 3.6674 0.0021 3.3665 0.0019

2R 0.9990 0.8950 0.4327

Notes: The dependent variable is the 25 firm-size category effects estimated by least squares controlling for labor forceŽ . Ž . Ž .experience through quartic , seniority, region, year, education eight categories and sex fully interacted . See Table III for

full results. The independent variables are the firm-size category averages for the indicated firm-specific compensationpolicy, adjusted for the same independent variables. The time period is 1976�1987.


TABLE IX

SUMMARY STATISTICS FOR FIRMS

ANNUAL AVERAGES OVER ALL YEARS FOR WHICH THE FIRM DOES BUSINESS 1978�1988Ž .weighted by inverse sampling probability

Variable Definition Mean Std Dev

Order-Independent Estimates

Ž .Average Predicted Effect of x Variables x� at the Firm 0.3569 0.2586Ž .Average Individual Effect, Unobserved Factors � at the Firm �0.0575 0.6626

Ž .Average Education Effect u� of Employees at the Firm 3.8889 0.2757�, Firm Effect Intercept �0.1791 1.0279� , Firm Effect Seniority Slope 0.0156 0.1167

Order-Dependent Estimates: Persons First

Ž .Average Predicted Effect of x Variables x� at the Firm 0.3906 0.2420Ž .Average Individual Effect, Unobserved Factors � at the Firm �0.0549 0.6446

Ž .Average Education Effect u� of Employees at the Firm 3.8503 0.2836�, Firm Effect Intercept �0.0196 0.2707� , Firm Effect Seniority Slope 0.0027 0.0775� , Firm Effect Change in Slope at 10 Years �0.0031 0.17282

Other Firm Characteristics

Number of Employees Sampled at Firm 34.2950 610.4800Ž .Employment at December 31st thousands 0.1097 1.6789

Ž .Real Total Assets millions FF 1980 59.4769 3,938.9800Operating IncomeTotal Assets 0.1254 0.4544Value AddedTotal Assets 1.0051 1.8889

Ž .Real Total Compensation millions FF 1980 1.3260 2.3570Ž .Real Value AddedEmployee thou. FF 1980 106.7672 936.5212Ž .Real Total AssetsEmployee thou. FF 1980 363.0707 21,067.5500

Ž .Engineers, Professionals and Managers Employee 0.2362 0.4072Skilled WorkersEmployee 0.5414 0.5255

Ž .Log Real Total Assets 1.7711 3.3558Ž .Log Real Value AddedEmployee 4.5215 1.1050Ž .Log Real SalesEmployee 5.5673 2.0139Ž .Log Total Employment at December 31 �3.0262 2.1109Ž .Log Real Total AssetsEmployee 4.7972 2.2710

Ž .Age of Firm N�7,385 19.5023 23.0331Number of Firms 14,717

Ž .Notes: Order-independent estimates are based upon x � , � , u� estimated with persons first conditional on Z and � andŽ . Ž .� estimated with firms first conditional on Z . Order dependent estimates are all based upon persons first x � , � , and u�

Ž .and firms � , � , and � second.2

ordinary least squares analysis in that much less of the variance is explainedŽ .between 0.30 and 0.55 . To assess the quality of the different methods inestimating the � coefficients of the time-varying observable personal character-

Ž .istics, we used Hausman 1978 tests to compare the coefficients obtained fromthe different methods with those obtained using the consistent method. Onceagain, all methods that include a person effect�the conditional method with


TABLE X

GENERALIZED LEAST SQUARES ESTIMATES OF THE RELATION BETWEEN

PRODUCTIVITY, PROFITABILITY AND COMPENSATION POLICIES

Ž . Ž .Dependent Variable: Log VAddedWorker Log SalesEmployee Operating Inc.Capital



Ž . Ž . Ž .Average Predicted Effect of 0.4937 0.0270 0.3050 0.0393 0.0670 0.0151Ž .x Variables x�

Ž . Ž . Ž . Ž .Average Individual Effect � 0.2234 0.0108 0.0809 0.0156 0.0081 0.0060Ž . Ž . Ž . Ž .Average Education Effect u� 0.1338 0.0254 �0.0057 0.0369 �0.0107 0.0143

Ž . Ž . Ž .�, Firm Effect Intercept 0.0371 0.0084 0.0054 0.0122 0.0138 0.0047Ž . Ž . Ž .� , Firm Effect Seniority Slope �0.1210 0.0582 �0.1751 0.0847 �0.0028 0.0328

Ž Ž . Ž . Ž .Engineers, Professionals, 0.3428 0.0238 0.1773 0.0346 �0.1303 0.0126.Managers Employee

Ž . Ž . Ž . Ž .Skilled Workers Employee 0.1226 0.0177 0.3065 0.0257 0.0061 0.0099Ž . Ž . Ž .Log CapitalEmployee 0.2470 0.0037 0.5536 0.0054

Ž . Ž . Ž .Intercept 3.0206 0.1055 0.1065 0.1533 0.1897 0.0579


Ž . Ž . Ž .Average Predicted Effect of 0.6057 0.0310 0.4833 0.0494 0.0569 0.0161Ž .Variables x�

Ž . Ž . Ž . Ž .Average Individual Effect � 0.2617 0.0118 0.1623 0.0188 0.0102 0.0061Ž . Ž . Ž . Ž .Average Education Effect u� 0.0725 0.0275 �0.0674 0.0437 �0.0036 0.0143

Ž . Ž . Ž .�, Firm Effect Intercept 0.1240 0.0343 0.1128 0.0546 0.0415 0.0179Ž . Ž . Ž .� , Firm Effect Seniority Slope 0.1492 0.1195 0.2852 0.1902 0.0571 0.0623Ž . Ž . Ž .� , Firm Effect Change in Slope �0.0485 0.0428 �0.1107 0.0681 �0.0264 0.02232

at 10 YearsŽ . Ž . Ž . Ž .Engineers, Tech., Managers 0.6815 0.0247 0.8989 0.0394 �0.1267 0.0126

EmployeeŽ . Ž . Ž . Ž .Skilled Workers Employee 0.2167 0.0190 0.4979 0.0302 0.0094 0.0099

Ž . Ž . Ž .Log CapitalEmployee 0.1017 0.0025 0.2290 0.0039Ž . Ž . Ž .Intercept 4.3985 0.1126 2.9784 0.1791 0.1664 0.0586

Note: Models were estimated using 14,717 firms with complete data. All regressions include a set of 2-digit industryeffects.

persons first or order independent, the within-persons method, and the within-persons with limited firm effects method�perform better than methods withoutperson effects.35 Both ordinary least squares and within-firm estimates yieldmuch higher � 2 statistics, indicating that our preferred model must includeperson effects that are not orthogonal to the time varying effects in the model,including the conditioning effects Z�.

35 All the � 2 statistics in models with person effects are around 3,500. In all cases, the statistichas 28 degrees of freedom. Hence, none of these models pass the test according to classical criteria.The models are also rejected using a simple Bayes-Schwartz criterion. However, given the largenumber of observations, we are likely to reject any model using these criteria. Hence, we use thistest statistic as a measure of proximity of the � estimates to the consistent ones.


TABLE XI

GENERALIZED LEAST SQUARES ESTIMATES OF THE RELATION BETWEEN

FACTORS USE AND COMPENSATION POLICIES

Dependent Variable

Ž ŽLog Real Log Capital EPM Skilled W Unskilled WŽ . . .Independent Variable Log Employees Capital Employee Employee Employee Employee


Average Predicted 0.2586 1.0369 0.7783 0.1420 0.0542 �0.1962Ž . Ž . Ž . Ž . Ž . Ž . Ž .Effect of x x� 0.0675 0.0971 0.0600 0.0110 0.0140 0.0132

Average Individual 0.2967 0.7673 0.4705 0.1197 �0.0284 �0.0913Ž . Ž . Ž . Ž . Ž . Ž . Ž .Effect � 0.0267 0.0384 0.0237 0.0043 0.0054 0.0051

Average Education 0.4380 0.5479 0.1100 0.2974 �0.1060 �0.1915Ž . Ž . Ž . Ž . Ž . Ž . Ž .Effect u� 0.0638 0.0918 0.0567 0.0102 0.0130 0.0123

�, Firm Effect Intercept �0.2654 �0.2898 �0.0244 0.0315 0.0152 �0.0468Ž . Ž . Ž . Ž . Ž . Ž .0.0212 0.0304 0.0188 0.0035 0.0044 0.0042

� , Firm Effect Seniority 0.4305 0.4149 �0.0156 0.0909 �0.0147 �0.0762Ž . Ž . Ž . Ž . Ž . Ž .Slope 0.1465 0.2106 0.1300 0.0241 0.0306 0.0290

Ž .Eng., Prof., Managers �0.0479 2.0645 2.1123Ž . Ž . Ž .Employee 0.0565 0.0812 0.0501

Ž .Skilled Workers �0.2505 0.1075 0.3580Ž . Ž . Ž .Employee 0.0444 0.0638 0.0394

Intercept �3.6868 2.6123 6.2991 �0.7097 0.8567 0.8530Ž . Ž . Ž . Ž . Ž . Ž .0.2587 0.3719 0.2296 0.0420 0.0534 0.0506


Average Predicted 0.2541 1.0205 0.7665 0.1142 0.0628 �0.1770Ž . Ž . Ž . Ž . Ž . Ž . Ž .Effect of x x� 0.0724 0.1036 0.0638 0.0117 0.0150 0.0142

Average Individual 0.2764 0.7454 0.4690 0.1231 �0.0316 �0.0914Ž . Ž . Ž . Ž . Ž . Ž . Ž .Effect � 0.0273 0.0391 0.0241 0.0043 0.0055 0.0052

Average Education 0.3478 0.4076 0.0598 0.3307 �0.0964 �0.2343Ž . Ž . Ž . Ž . Ž . Ž . Ž .Effect u� 0.0643 0.0921 0.0567 0.0101 0.0129 0.0122

�, Firm Effect Intercept 0.3748 0.7618 0.3869 0.0057 �0.0052 �0.0005Ž . Ž . Ž . Ž . Ž . Ž .0.0802 0.1148 0.0707 0.0131 0.0167 0.0158

� , Firm Effect Seniority �0.0262 0.5277 0.5539 0.0835 �0.0303 �0.0532Ž . Ž . Ž . Ž . Ž . Ž .Slope 0.2798 0.4005 0.2467 0.0456 0.0582 0.0553

� , Firm Effect Change 0.0011 0.0497 0.0486 �0.0314 0.0140 0.01742Ž . Ž . Ž . Ž . Ž . Ž .in Slope at 10 Years 0.1002 0.1435 0.0884 0.0164 0.0209 0.0198

Ž .Engi., Prof., Managers �0.1181 2.0038 2.1219Ž . Ž . Ž .Employee 0.0568 0.0812 0.0500

Ž .Skilled Workers �0.2947 0.0707 0.3654Ž . Ž . Ž .Employee 0.0445 0.0637 0.0392

Intercept �3.4129 3.0371 6.4499 �0.8485 0.8309 1.0176Ž . Ž . Ž . Ž . Ž . Ž .0.2630 0.3765 0.2319 0.0423 0.0539 0.0512

Notes: The models were estimated using the 14,717 firms with complete data. All equations include a set of 2-digitindustry effects. Standard errors in parentheses.


TABLE XII

PROPORTIONAL HAZARDS ESTIMATES OF THE RELATION BETWEEN FIRM SURVIVAL AND

COMPENSATION POLICIES

Parameter Standard RiskIndependent Variable Estimate Error Ratio


Ž . Ž .Average Predicted Effect of x x� 2.2163 0.5821 9.1730Ž . Ž .Average Individual Effect � �0.5874 0.2100 0.5560Ž . Ž .Average Education Effect u� �2.3441 0.5327 0.0960

Ž .�, Firm Effect Intercept 0.3833 0.1579 1.4670Ž .� , Firm Effect Seniority Slope 1.2239 1.0215 3.4000

Ž . Ž .Eng., Prof., Managers Employee 0.2328 0.3689 1.2620Ž . Ž .Skilled Workers Employee 0.2065 0.2917 1.2290


Ž . Ž .Average Predicted Effect of x x� 2.0751 0.6241 7.9650Ž . Ž .Average Individual Effect � �0.5327 0.2064 0.5870Ž . Ž .Average Education Effect u� �1.8615 0.5398 0.1550

Ž .�, Firm Effect Intercept �0.5909 0.5356 0.5540Ž .� , Firm Effect Seniority Slope 1.6497 2.4598 5.2050Ž .� , Firm Effect Change in Slope at 10 Years 0.3592 0.6677 1.43202

Ž . Ž .Eng., Prof., Managers Employee 0.4096 0.3699 1.5060Ž . Ž .Skilled Workers Employee 0.3372 0.2926 1.4010

Notes: Negative coefficients indicate a reduced probability of firm death. This model was estimated using the 7,382 firmswith known birth dates. The model includes a set of 2-digit industry effects.

Ž .We also computed the specification test shown in equation 3.40 , which teststhe hypothesis that the conditioning variables Z used to compute the columnlabelled ‘‘Conditional Method Persons First’’ are adequate to represent thecovariance between personal characteristics, both measured and unmeasured,and firm effects. The computed statistic is 21,000 with 48 degrees of freedom.Since the conditioning variables have the most effect on the results when firmeffects are estimated using the order-independent method, we conclude that thislarge � 2 statistic, in conjunction with the component correlation analysis wediscuss below, is evidence that the order-independent estimated firm effects areless reliable than the firm effects from the order-dependent ‘‘persons first’’method. Of course, with the large sample sizes in this analysis, it is also the casethat the large value of this statistic can be interpreted as having enough data to

Ž .reject unsurprisingly a low-dimensional simplification of the covariance be-tween X, D, and F. In spite of the data evidence that one should permit all ofthe effects to be correlated and that one should estimate person effects first inthe conditional method, we present all of our results using both the order-independent method and the order dependent ‘‘persons first’’ methods. None ofour conclusions are affected by our choice of conditional estimator.


5.2.2. Male-female wage differentials

Comparing the results for men and women in Table III, we note that there isˆ 36less variation in the � across estimation methods for women than for men. We

also note that the gender gap is decreasing over our period of analysis accordingto the least squares estimates of the time effects with no person or firm effects.However, changes in the composition of the work force must have been animportant determinant of this trend because, when person effects are included,the estimates of the time effects are virtually identical for the two sexes. Thus,given the overall difference between men and women in the French labormarket, once we control for personal heterogeneity, there is no evidence ofdeclining or increasing male-female wage differentials. As usual, the experienceprofile for women is flatter than for men, regardless of the method of estima-tion, even though, for our data source, the measure of labor force experienceexcludes within-sample periods of nonemployment.

5.3. Discussion of the Estimated Person and Firm Effects

Table IV contains descriptive statistics for the components of real compensa-tion implied by the estimated parameters from both of our conditional methodspecifications, estimated separately for each sex. Table V contains pooledsummary statistics and corrected correlations for all of the components of real

Žcompensation and for two different conditional estimation methods order.independent and order dependent ‘‘persons first’’ . The table also contains the

different estimates of the seniority effects based on the estimation techniquespresented in Table III. For both males and females, the standard deviations ofthe individual-effect, � , and its components � and u�, are the same order ofmagnitude as the firm effects, � , for the order-independent method andsubstantially larger than those of the firm effects for the order-dependentmethod with persons first. As noted in Table III, the complete parameterizationexplains about 80% of the variation in real annualized earnings; thus, theperson-specific component of variance is clearly important. The firm-specificcomponent of variance is less important but still a major source of variation inthe compensation data.

5.3.1. Specification checks based on correlations among theheterogeneity components

To further compare the different estimation methodologies, Table V showsthe correlations among the components of person and firm heterogeneity as

Ž .estimated using order-independent, order-dependent both ways , within persons

36 This statement is based upon the average variation in the coefficients for men versus those forwomen from the estimates in columns labelled ‘‘Consistent Method Person and Firm Effects,’’‘‘Conditional Method Persons First,’’ ‘‘Within Persons No Firm Effects,’’ ‘‘Within Persons LimitedFirm Effects,’’ and ‘‘Within Firms No Person Effects.’’


with firm specific intercepts and seniority slopes for the 115 largest firms, andconsistent methods. This table is particularly complicated and some care isrequired to read it properly. The correlation coefficients reported in the tableare all computed to be representative of persons; hence, we use the full sampleof 5,305,108 observations for all methods except for correlations with the fullleast squares estimates with limited firm effects, where the number of observa-tions is equal to the 695,077 person-years for which the firm coefficients areavailable.

The panel labelled ‘‘Firm Effects’’ contains correlations of the components ofthe firm effects, � and � , by method of estimation. In the ‘‘Firm Effects’’ panel

Ž Ž ..the order-independent estimates based on equation 3.16 are conditional onZ but exclude person effects; hence, they are equivalent to order-dependent‘‘firms first’’ estimates, conditional on Z. In this same panel the order-depen-dent estimates are persons first, conditional on Z. The full least squaresestimates using the 115 largest firms show the firm effects from the appropriateequation reported in Table III. Finally, in the ‘‘Firm Effects’’ panel, the

Ž .consistent estimates are based on equation 3.8 . Note that consistent estimatesof the � component of the firm effect are not available.

In the panel labelled ‘‘Person Effects,’’ we report correlation coefficientsŽ .based upon the order-independent estimates in equation 3.15 , which are

equivalent to order-dependent estimates with persons first. In this same panelwe report person effect estimates from the order-dependent method with firmsfirst.37 Finally, the person effects from the model labelled ‘‘Full Least SquaresEstimates Using the 115 Largest Firms,’’ are based on the estimates reported inTable III. Note that consistent estimates of the person effects are not available.

5.3.2. Specification checks based on firm effects, including heterogeneousseniority slopes

Consider first the firm-specific intercept, �. The results in Table IV show that,for both sexes, the standard deviation of the estimated firm effects is very large.The data evidence that the complete firm effect, � , is heterogeneous is particu-larly compelling when we combine the results shown in Table IV with the formalspecification analyses we showed in Table III. Furthermore, the results in TableV show that order-dependent and order-independent methods give very differ-ent results for the firm effect since the correlation between the two estimates of� is only 0.16. For the subsample of persons employed in the largest firms, thefull least squares solution for � appears to be closest to the consistent methodŽ .see below . Thus, we assess the quality of the estimates of � obtained by theorder-dependent and order-independent methods by comparing them with thefull least squares solution for this subsample. Using this criterion, � as esti-

37 Because, as the reader will see shortly, these estimates perform very poorly, we do not reportany other estimates based on the order-dependent method with firms first.


mated by the order-independent method is weakly correlated with � as esti-mated with limited firm effects. On the other hand, the � estimated by theorder-dependent ‘‘persons first’’ method is strongly correlated with the � esti-

Ž .mated with limited firm effects correlation of 0.67 . Thus, the evidence basedon � favors the order-dependent ‘‘persons first’’ conditional estimation method.For clarity we stress that both conditional methods imply very substantial firmeffects. The conclusion from this specification discussion is that the similarity

Ž .between the full least squares estimates of � for the 115 largest firms and theorder-dependent ‘‘persons first’’ estimates indicates that the order-independentestimates of � confound the pure firm-specific intercept with the average personeffect within the firm.

Considering next the seniority coefficients, � , all methods in which we allowthese returns to vary across firms show that the standard deviation of theestimated seniority slopes is large, at least three times the mean. Our results,therefore, strongly suggest that earnings equations should have a firm effectwith at least a firm-specific intercept and seniority slope. The various estimationmethods, however, also show considerable variability in � across techniques.The average seniority coefficient is about 0.01 whenever the estimation method

Ž 38excludes person effects the order-independent method, ordinary least squares,.within firms, within industry and within size class . The average seniority

Žcoefficient decreases to near zero when person effects are included order-de-.pendent persons first, within persons, and 115 largest firms . The consistent

method, which includes person effects, gives results closer to the models thatexclude person effects�around 0.01 for the average seniority slope.

To continue our discussion of the seniority effects, consider the correlationamong our estimates of this component of firm heterogeneity. Because we haveconsistent estimates of the seniority coefficients, it is useful to examine thecorrelation of � from the consistent method with the other estimates. First, thecorrelation of the consistent � with the � estimated in the order-independent

Ž .method is quite low 0.09 . The correlation of the consistent � with the oneestimated by the order-dependent ‘‘persons first’’ method is only slightly larger.However, on the restricted sample of individuals for which the full least squaressolution has been implemented, the correlation of the consistent � with the full

Ž .least squares � is quite high 0.51 . In fact, the � estimated using the full leastsquares solution with the 115 largest firms is well-correlated with all of themethods of recovering � over the subsample for which this estimate is available.Hence, for the largest firms, the seniority slope coefficients seem to be reason-ably estimated by any of our methods. However, for the other, smaller, firms, noestimation method appears to dominate in a clear-cut fashion if one relies onlyon � to assess the methodology.

38 In the order-independent method the firm effects are estimated without first eliminating personeffects; thus, they exclude person effects.


5.3.3. Implications of heterogenous seniority slopes

As we noted in Section 2, by considering the possibility of differential returnsto seniority as a part of the firm effect, we can provide some direct evidence onthe debate surrounding the interpretation of the average seniority effect. Usingour consistent estimates of the return to seniority, � , we find that the averagereturn to a year of seniority is just over 0.01 for both sexes. This estimate is

Ž . Ž .lower than Topel’s 1991 result but consistent with Brown’s 1989 results whenhe includes person effects. The heterogeneity in our consistent estimates sug-gests that some of the difference between our results and Topel’s may be due tocorrelation between the heterogenous firm effect and the person effects. Thefact that our results are closer to Brown’s supports this conclusion becauseBrown’s seniority effect is heterogenous�the magnitude of the return toseniority depends upon characteristics of the job�and he permits correlationbetween this heterogeneity and his person effect. Brown, on the other hand,does not allow for the possibility of firm-specific intercepts, except as reflected inthe job characteristics he used to model the heterogeneity in the return toseniority. Although we cannot use our consistent technique to address thisquestion, we note that, for all the preferred estimates of � , there is a negativecorrelation between � and the associated estimate of �. This negative correla-tion indicates that the firm-specific intercept and the firm-specific seniority slope

Ž .are negatively correlated, a result predicted by Becker and Stigler 1974 andŽ .Lazear 1979 .

5.3.4. Specification checks based on person effects

Consider now the correlation between the different estimates of � . Anargument similar to the analysis we used for � shows that the � ’s estimatedwith persons first are better than those estimated with firms first. In theestimation of � , the order-independent estimates are mathematically identicalto the order-dependent ‘‘persons first’’ estimates, conditional on Z. The alterna-tive method is to consider the order-dependent ‘‘firms first’’ estimation of � . Wenote that the correlation between the order-independent estimates and the fullleast squares solution for the 115 largest firms is 0.99, while the order-dependent‘‘firms first’’ estimates are only correlated 0.58 with the order-independentestimates and 0.60 with the full least squares solution for the 115 largest firms.These correlations indicate that the order-dependent: ‘‘firms first’’ estimates of� are not capturing the pure person effect as reliably as either of the other twoalternatives shown in the ‘‘Person Effects’’ panel of Table V.

5.3.5. Implications of the correlations among compensation components

Table VI shows the intercorrelations of the different components of compen-sation, first for the order-independent method, then for the order-dependent‘‘persons first’’ method. Both methods indicate that � , the unobservable part of


the individual effect, is the component of compensation that is most highlyŽcorrelated with log real annual total compensation 0.80 or 0.73 depending on

. 39the method . The firm components are much less important in the determina-Ž .tion of total compensation 0.21 or 0.26 depending on the method . Using the

order-independent estimates, the � component of the person effect and the �component of the firm effect are positively correlated 0.15. The estimatedcorrelation is 0.08 using the order-dependent ‘‘persons first’’ estimates. In eithercase, the estimated correlation between firm and personal heterogeneity is notlarge. Also notice that, although the firm-specific intercept, �, and the �-compo-nent of the person effect are positively correlated, the firm-specific intercept is

Žnegatively correlated with the seniority slope �0.07 order independent and.�0.56 order dependent ‘‘persons first’’ . In both methods, the correlation

between observables and compensation appears to be smaller than the correla-tion between unobservables and compensation. The correlation between com-pensation and education, u �, is around 0.4 and the correlation betweenicompensation and the time-varying individual characteristics, x � , is aroundi t0.3, for both methods shown in the table. Furthermore, x � is only weaklyi tnegatively correlated with the unobservable � .40

5.3.6. Summary of the eidence from the estimation results on person andfirm heterogeneity

After reviewing the evidence of the quality of the different estimationmethods, the following conclusions can be drawn. First, person effects tend to bemore important than firm effects in explaining compensation variability. For theparameters � and � , the estimation methods with persons first are preferred bythe data. However, there is no definitive evidence in favor of one estimationmethod over another for the firm effect � . On one hand, the order-dependent

Ž .‘‘persons first’’ method tends to give results on � that are more highlycorrelated with the consistent estimates. On the other hand, the order-indepen-dent method produces estimates of � that are less correlated with thoseobtained by the consistent methodology. Hence, in what follows, we will examinethe classical problems of labor economics that were mentioned in the motivation

Žsection using person effects that have been estimated first i.e. person effects.from the order-independent and order-dependent ‘‘persons first’’ methods and

firm effects from these same methods, in all cases conditional on Z. Thisprovides us with results that reflect the widest range of possibilities regardingthe appropriate estimate of the firm effect.

39 As noted in the discussion of statistical methods, at the level of the individual the least squaresestimate of the person effect is unbiased but inconsistent. Thus, the variance of � as directlyî

� � � �calculated from the summary measures consists of two components var � � var � �� , andî i isimilarly for � . The variances used to calculate all correlations with � and � in Table VI havei i i

ˆ� � � �been corrected by subtracting an estimate of var � �� , var � �� , respectively.î i i i40 Recall that � is orthogonal to u � by construction.ˆ î i


5.4. Inter-Industry Wage Differentials

Ž .In Table VII, we implement the equation 2.7 derived in Section 2, whichallows us to decompose the industry effects, estimated as shown in Table III,into the component due to pure firm effects and the component due to personeffects. Notice that the two right-hand side components must be adjusted for the

Ž .observables as in equation 2.7 . Table VII uses industry-level averages of theindividual and firm-specific components of compensation to explain the industry

Žeffect found in our raw individual data taken from the regression controllingfor labor force experience, seniority, region, year, education, and sex reported in

.the column labelled ‘‘Within Industry No Person Effects’’ in Table III in theŽ .spirit of Dickens and Katz 1987 . Since the industry-average person and firm

effects, also adjusted for the same set of factors as reported in the Table IIIregression, almost fully account for the industry effects in a statistical senseŽ 2R �0.85 using person and firm effects drawn from the order-independentmethod and R2 �0.96 using measures drawn from the order-dependent method

.with persons first , the interesting question concerns the relative importance ofŽ .individual heterogeneity the �-component of the person effect, in particularŽ .and firm heterogeneity the �-component as components of the industry

effects. For both estimation methods for the firm effects, the separate influenceof person effects and firm effects in explaining the industry effects is shown. Theseparate analyses confirm the relative importance of person, as compared tofirm, effects. The third through sixth columns of Table VII present separate

Žindustry-level regressions using, first, industry-average � alone columns 3 and. Ž .4 and, then, using industry-average firm effects alone columns 5 and 6 . It is

Žclear from the fact that industry-average � alone explains 84% 92% with the.order-dependent estimates with persons first of the inter-industry wage varia-

Žtion, whereas the industry-average � component explains only 7% 25% with.the order-dependent estimates , that individual effects, as measured statistically

by � , are more important than firm-components, as measured by � , for explain-ing French inter-industry wage differentials.41

Figures 1 and 2 show graphically the important difference in the strength ofthe relation between industry effects and industry-average person and firm

Ž .effects. Figure 1 plots the industry effects from equation 2.7 , the dependentvariable in Table VII, against the industry-average person effects. The figurealso shows the fitted regression line. Figure 2 plots the same industry effectsagainst the industry-average firm effects, again showing the fitted regressionline. Both figures are based on the order-independent estimates. The relationbetween the raw industry effects and the industry-average person effects isclearly much stronger than the one between raw industry effects and theindustry-average firm effect. The graphical display for the order-dependent‘‘persons first’’ estimates shows the same results.

41 As shown in Table VI, these two components are not highly correlated, so little of theindustry-average person effect is ‘‘explained by’’ the industry-average firm effect in a statisticalsense.


FIGURE 1.�Actual and predicted industry effects using industry-average person effects.

FIGURE 2.�Actual and predicted industry effects using industry-average firm effects.


5.5. Firm-Size Wage Effects

Table VIII presents a similar analysis for the firm-size effect based onŽ .equation 2.9 . To implement this analysis, we constructed 25 firm-size cate-

gories. We then estimated the firm-size effects without controlling for person orŽ .firm effects, as in equation 2.9 . Using calculations exactly parallel to those in

Table VII, we constructed the appropriate weighted average person and firmeffects within each firm-size category, conditional on the same X variables usedin the other analyses. The complete set of X coefficients is shown in Table III inthe column labelled ‘‘Within Firm Size No Person Effects.’’ Table VIII showsthat, for both methods of estimating the person and firm effects, the firm-size-average person effect is much better at explaining the firm-size wage effect thanis the firm-size-average firm effect.42 To more easily compare our results to

Ž .others, Brown and Medoff 1989 in particular, we graph the raw firm-size wageeffects against the log of firm size in Figure 3. The raw firm-size effects in ourdata strongly resemble the effects summarized by Brown and Medoff. The

Ž .relation between firm size log of employment at the firm and compensation,controlling for the observable characteristics, follows a concave quadratic rela-

Ž .tion. Figure 3 also plots the average person effect hollow boxes within firm-sizecategory. The average person effect can be seen to follow essentially the samequadratic function of log firm size and many of the average person effects arecoincident with the solid dots representing the raw firm-size effect. Finally,

Ž .Figure 3 shows the average firm effect hollow triangles . The average firmeffects do not follow the same concave quadratic function of log firm size as theother two effects. Indeed, the relation between the firm-size average firm effectand log firm size is slightly convex, with the largest positive average firm effectoccurring in the largest firm-size category and the largest negative average firmeffect occurring in the second largest firm size category. The effects plottedclearly show that average person effects are much more closely related to thefirm size effects than average firm effects. The results shown are based on theorder-independent estimates but are essentially the same for the order depen-dent estimates-persons first.

From these two analyses, we conclude that person effects are much moreimportant in explaining inter-industry wage differentials and firm-size wageeffects.

5.6. The Economics of Human Resource Management

We turn now to the analysis of the impact of the compensation structure onfirm outcomes. To conduct this analysis we first computed the firm average ofthe different components of the compensation package as measured by ourorder-independent and order-dependent ‘‘persons first’’ methods. Hence, we

42 As in the analysis of Table VII, the size effects used for the analysis in Table VIII come fromthe column in Table III labelled ‘‘Within Firm Size No Person Effects’’ and the size-class averageperson and firm effects have been adjusted for the same effects as found in the Table III regression.


FIGURE 3.�Firm size effects related to firm-size average person and firm effects.

computed the average for each firm j of the part of compensation due toŽ . Ž .education u � , to time-varying observables x � , and to non-time-varyingi i t

Ž . Ž .unobservables � , using all observations i, t for which individual i wasiworking in firm j at date t. The detailed formulas for this computation aredescribed in the model section and the variables available for study are de-scribed in the data description.

ŽTable IX presents summary statistics for the sample of firms weighted to be.representative of private industrial firms . Table X presents regression models of

the logarithm of real value added per employee, real sales per employeeŽ .measures of productivity , and operating income as a proportion of total assetsŽ .a measure of performance . Results are reported for the order-independentand the order-dependent ‘‘persons first’’ methods. Using the firm-level compen-sation policy measures generated by our methods, we note that a larger value ofthe average component of the wage associated with time-varying characteristicsŽ .x� is associated with higher value-added and sales per worker and higherprofitability for both estimation methods. A larger firm-average individual effectŽ .� is associated with a substantially larger value-added per employee and salesper employee but not with higher profitability. Once more, these results areconsistent across estimation techniques. The part of the individual-effect related

Ž .to education u� is associated with higher value-added per worker but is notsignificant in the other two columns, irrespective of the estimation method.

Ž . ŽHigher firm-specific wages � are associated with higher productivity value-ad-


ded per worker and sales per worker, albeit not with the order-independent.method for this last variable and with higher profitability.

The differences between the results based on the order-independent andorder-dependent estimation methods, as shown in Table X, are most striking

Ž .when looking at the impact of the seniority slope coefficient � . Using theŽorder-dependent estimates, neither seniority slope is associated with higher or

.lower productivity or profitability. However, using the estimates from theorder-independent method, it appears that there exists a negative associationwith firm productivity�firms that reward seniority the most tend to be the leastproductive.

The results in Table X can also be used to discuss the relation between firmlevel compensation policies and measurable outcomes in the context of hiring,rent-splitting, and efficiency wage models. Individuals with high opportunitywages, as captured by � , tend to work in firms with higher productivity perworker, as measured by either value-added per worker or sales per employee.Recall that the �-component of personal heterogeneity has been estimatedusing compensation as the dependent variable. Thus, it represents the market’svaluation of this personal heterogeneity. It is thus not surprising that there is noprofitability effect associated with � ; however, for the same reason, the pres-

Ž .ence of an association between the observable characteristic component x� ofcompensation and profitability is puzzling, especially since the education compo-nent of individual heterogeneity has no measured association with profitability.The firm-specific effect in compensation, as measured by the firm-specific

Žintercept �, is associated with both higher productivity value-added for either.method and sales for the order-dependent measure and higher profitability.

This result can be interpreted as evidence consistent with some efficiency wageor rent-splitting activity in the labor market.

Table XI presents the results for the relations among our compensationmeasures and a variety of firm-level factor utilization rates. Results are alsoreported for both conditional estimation techniques. Larger values of thefirm-average, time-varying component of compensation, x� , are associated withhigher employment, capital, capital-labor ratio, proportion professional employ-ment, and proportion skilled employment and with lower unskilled employment.The unobservable component of the individual effect, � , is positively associatedwith employment, capital, the capital-labor ratio, and the proportion of engi-neers, technical workers, and managers in the work force; and is negativelyrelated to the shares of both skilled and unskilled workers. Larger values of the

Ž .average education effect the observable component of the individual effect, u�are associated with higher employment, capital, and proportion professionalsbut with lower values of the proportion skilled. All of these results holdregardless of the estimation method.

The estimation method for the compensation components matters whenexamining the impact of the firm effects on these outcomes. Based on theorder-dependent ‘‘persons first’’ method, the firm-specific intercept, �, is stronglypositively associated with employment, capital, and the capital-labor ratio; but is


not associated with any components of the skill structure of the work force. Ahigh firm-specific seniority slope is positively associated with the capital-laborratio and slightly positively associated with the proportion of professionalemployees. Based on the order-independent method, all of the associations with� that were positive using the order-dependent method are now negativeŽ .significantly for employment and capital, marginally for the capital-labor ratio ;but the firm-specific seniority slope, � , plays the role that the firm-specificintercept, �, played with the other estimation method�it is positively associ-ated with employment and capital. Furthermore, managerial and skilled employ-ment are both positively associated with firm-specific effects. It appears that ourtwo estimation techniques both capture similar effects, but their allocation tothe fixed part and to the seniority part of firm-specific heterogeneity differ. Thisis confirmed by a look at Table V in which we see that � from the order-depen-dent method is highly negatively correlated with � from the order-independentmethod and that the � from the order-independent method is somewhatnegatively correlated with � from the order-dependent method.

Finally, Table XII presents a proportional hazards analysis of the relationbetween the survival of firms and our estimated compensation components atthe firm level.43 Both components of the individual effect, � and u�, areassociated with an increase in survival probability in a statistically significantmanner. The effects related to firm-specific compensation factors are large butvery imprecise, even though a high � tends to decrease survival when usingorder-independent estimates. The effect associated with the firm average ofobservable personal characteristics, x� , is also associated with a decreasedsurvival probability. The results are interesting when combined with those foundin Table X. High � is related to high profitability with both estimation methods,but is linked to lower probabilities of firm survival. On the other hand, high � isrelated to increased survival probabilities, but has no significant relation toprofitability.

6. CONCLUSIONS

In Section 2 we identified six broad areas of labor economics that could beadvanced by the study of matched longitudinal employer-employee data:

� the role of individual and firm heterogeneity in the determination of wagerates;

� the sources of inter-industry wage differentials;� the sources of firm-size wage effects;� the role of seniority, and heterogeneous returns to seniority in determining

wage rates;� the measurement of internal and external wage rates;� the study of the economics of human resource management policies.

43 We estimate the Cox proportional hazards model using as independent variables the non-time-varying measures shown in Table XII. The nonparametric baseline hazard was not estimated.


We believe that our analysis of the French compensation data, linked to theeconomic performance data of the employing firms has, indeed, shed consider-able new light on these questions. To summarize, we found:

� Personal heterogeneity and firm heterogeneity were both important deter-minants of compensation, although personal heterogeneity appears to be sub-stantially more important in these French data.

� Across 84 industries, the industry-average person effect, adjusted for inter-industry differences in observable characteristics, is much more important thanthe industry-average firm effect, similarly adjusted, for explaining the inter-industry wage differential.

� Across 25 employment-size categories, the firm-size wage effect in France isincreasing at a decreasing rate and this effect is more closely predicted by asimilar pattern in the firm-size-average person effect than by the firm-size-aver-age firm effect, which does not mirror the raw firm-size effects at all.

� There is considerable evidence for heterogenous returns to seniority but themethod of estimating the return to seniority affected the conclusion regardingthe average return to one year of additional seniority. Returns to seniority arenegatively correlated with firm-specific intercepts in the compensation relation.

� If we associate the person effect with an individual’s external wage rate andthe firm effect with that person’s internal wage rate, there is very little correla-tion between these two measures, suggesting that models that focus on explana-

Ž .tions for the individual heterogeneity human capital and models that focus onŽexplanations for the firm heterogeneity compensation design, incentives, bar-

.gaining are addressing features of the labor market that do not have largeinteractions.

� Firms that hire ‘‘high-wage workers,’’ those with above average personeffects, are observed to have more productive work forces but no higherprofitability. ‘‘High-wage firms,’’ those that pay above average firm effects, areobserved to have both more productive work forces and higher profits.

Of course, our analysis of the separate effects of individual and firm hetero-geneity on wage rates and on firm compensation policies has also raised manynew questions:

� Do the results for France generalize to other labor markets?� If person effects are much more important than firm effects in explaining

variation in compensation, do these same effects also explain employmentmobility?

� If pure firm effects are not very important in the explanation of inter-in-dustry wage differences, then why do other analyses that control for personalheterogeneity but not for firm heterogeneity appear to suggest otherwise?

� Does the observed relation between hiring ‘‘high-wage’’ workers and havinghigher productivity per worker mean that employer’s hiring and selectionmethods should be studied more closely?

� Is the observed relation between being a ‘‘high-wage’’ firm and being bothmore profitable and more productive per worker evidence that efficiency wagemodels play a role in explaining inter-firm differences in compensation policies?


Although we have provided considerable new evidence on these outstandingquestions, we believe that our results also provide the statistical basis uponwhich to begin the process of testing the relevance of agency, efficiency wage,searchmatching, rent sharing and endogeneous mobility models as potentialexplanations for compensation outcome heterogeneity.

Dept. of Labor Economics, 259 Ies Hall, Cornell Uniersity, Ithaca, New York14853-3901, U.S.A.; [email protected]; http:old-instruct1.cit.cornell.edu:8000abowd-john; CREST, Malakoff Cedex, France, and NBER, Cambridge,MA, U.S.A.,

CREST-INSEE, Dept. de la Recherche, 15, Bd. Gabriel Peri, 92245 MalakoffĆedex, France; [email protected]; and CEPR, London, U.K.,

and( )CNRS, LAMIA-TEAM, Uniersite de Paris 1 Pantheon-Sorbonne , Maison des´ ´

Sciences Economiques, 106-112, Bd. de l’Hopital, 75647 Paris Cedex 13, France;ˆ[email protected]; and CREST, Malakoff Cedex, France

Manuscript receied February, 1994; final reision receied January, 1998.

STATISTICAL APPENDIX

In this appendix we state and prove the basic statistical results relating our estimation techniquesand our analysis of the aggregation and suppression of effects to the standard least squares analysisof individual and firm effects that have been estimated in other contexts. The model is stated in

Ž .equation 2.2 and the definitions that follow. We use the same notation in this appendix.Ž .There are a total of J firms indexed by j�1, . . . , J. The function J i, t gives the identity of the

employer for individual i in period t. For each individual i and each year t�n , . . . , n , a row ofi1 iTi

the matrix F contains an indicator variable for which the jth column contains the value 1 and all0Ž . �other columns contain the value 0, where j�J i, t . The matrix F is, thus, N �J and the0

associated vector of firm effects, �, is J�1. A row of the matrix F contains, for each individual i1and each year t�n , . . . , n , in the jth column the value of the individual’s seniority in the firmi1 iTi

Ž .j�J i, t , s , and 0 in all other columns. A row of the matrix F contains, for each individual i andi t 2each year t�n , . . . , n , in the jth column the value of the individual’s seniority in the firmi1 iTi

Ž . Ž .j�J i,t less 10 if this value is positive and 0 otherwise, T s �10 , and 0 in all other columns. The1 i tcomplete firm effect can thus be represented as

F��F ��F ��F �0 1 2 2

� � � � � � ��where F� F F F and �� .0 1 2 2The error vector, � , is N� �1 and has the following properties:

� �E � �X , D , F �0,

� � 2�var � �X , D , F �� I .N

Hence, the full regression equation for the model in the main text of the paper is given by

Ž .7.1 y�X��D��F ��F � �F � �� .0 1 1 2 2

For completeness we note that the N individuals constitute a simple random sample of theŽ .population of persons ever employed outside the government sector between the years 1976 and

Ž1987 except for 1981 and 1983, for which the data were not made available in a computerized.sample . In general, the individuals were sampled if their birth dates fell in October of an even year.


Once sampled, an individual’s complete private-sector employment history between the years 1976and 1987 is available, again except for the years 1981 and 1983.

Ž .At most P�N� 3 J�1 effects in the full model are identified. The least squares estimator ofthe complete set of effects is given by

��ˆ � � �� X yX X X D X F�� Ž . D y7.2 �ˆ D X D D D F�

� � � �F X F D F F F y�

� ��where the notation represents any generalized inverse. The standard method of calculating theleast squares estimates of the effects is to take deviations from the within-person means of the

Ž .variables. This operation is accomplished by premultiplying both sides of equation 7.1 by the matrix� Ž � .�1 � �M � I�D D D D . The least squares estimator of the identifiable effects can, then, beD

restated as

�� ˆ X M X X M F X M y� D D DŽ .7.3 � .� � �F M X F M F F M yˆ D D D�

It is because the off-diagonal submatrix X �M F is neither null, patterned, nor sparse that weD˜Ž .cannot directly compute the solution 7.3 . Furthermore, even if we use the consistent estimators �

Ž . Ž .and � from equations 3.9 and 3.10 and set � �0, because of the presence of the person effects,˜ 2˜ Ž .the consistent estimator for � based upon equation 7.3 is

�� ˜ ˜Ž . Ž .�� F M F F M y�X��F � ,˜0 D 0 0 D 1

which still requires the solution of a system of J equations that is neither diagonal, patterned, norsparse.

Computation of the Conditional Estimates

Ž . Ž .The calculation of the solution to equations 3.17 and 3.18 for the order-independent andŽ .order-dependent persons first methods do not present any problems. The calculation of equation

Ž .3.21 for the order-independent and order-dependent ‘‘firms first’’ methods is, however, morecomplicated. For the order dependent method with firms first we leave F � in the model; however,2 2we do not attempt the order independent calculations with F � in the model.2 2

To calculate � , we reorganize the columns of F so that the columns of F and F from the sameˆ 0 1firm are adjacent. Next, we sort the matrix F so that the observations are grouped by firm fromj�1, . . . , J. Denote the reorganized F matrix by F� and denote the conformably reorganized y andZ matrices by y� and Z� , respectively. Note that the cross-product matrix F��F� is block diagonalwith J blocks, each one 2�2, and a typical block is given by

N sÝj i tŽ .J i , t � j

2s sÝ Ýi t i tŽ . Ž .J i , t � j J i , t � j

where

� Ž . �N � 1 J i , t � j ,ÝjŽ .� i , t

Ž . Ž . � �the notation Ý means to sum over all i, t such that J i, t � j, and the function 1 A �1 if AJŽ i, t .� jis true and 0, otherwise. Similarly, the cross-product matrix F��Z� , which is 2 J�Q, has the


structure

z zÝ ÝŽ i , t .1 Ž i , j.QŽ . Ž .J i , t � j J i , t � j

s z s zÝ Ýi t Ž i , t .1 i t Ž i , t .QŽ . Ž .J i , t � j J i , t � j

.

z zÝ ÝŽ i , t .1 Ž i , t .QŽ . Ž .J i , t � j J i , t � j

s z s zÝ Ýi t Ž i , t .1 i t Ž i , t .QŽ . Ž .J i , t � j J i , t � j

Ž �� .�1 �� The product F F F Z is, therefore, a 2 J�Q matrix of firm-specific regression coefficients.A similar argument can be made for the coefficients associated with the regression of y� on Z�.Hence, the adjustment of Z� with respect to F� can be accomplished performing firm by firmregression of the appropriate rows of each column of Z� on the appropriate columns of F� and

Ž �� . Ž �� .� �retaining the residuals to cumulate in the cross-product matrices Z M Z and Z M y .F FThus

�1�� Ž . Ž .� �� Z M Z Z M y ,ˆ F F

where we note that it is not necessary to adjust y� with respect to F� as long as each column of Z�

Ž .�has been adjusted i.e. the matrix M is idempotent .FA computationally identical approach to the estimation of � may be obtained by directly solving

Ž .the least squares equations associated with the solution of 3.20 . To begin, notice that the leastsquares solution to this equation has the property

� �y �� s �z � �0Ý i t j j i t i tŽ .J i , t � j

for j�1, . . . , J, where the variable z is a row of the matrix Z. These J conditions implyi t

ˆŽ .7.4 � �y �� s �z �ˆ ˆj j j j j

where the notation

Ý aJŽ i , t .� j i ta � .j � Ž . �Ý 1 J i , t � j� Ž i , t .

Next, consider the J orthogonality conditions associated with the variable s , which implyi t

ˆ 2y s �� s �� s �z s � �0ˆ ˆÝ Ž .i t i t j i t j i t i t i tŽ .J i , t � j

for j�1, . . . , J. Hence,

�Ž . Ž . �Ý y �y s � z �z s �JŽ i , t .� j i t j i t i t j i tŽ .7.5 � � .ˆj Ž .Ý s �s sJŽ i , t .� j i t j i t

Ž . Ž . Ž .Substituting equations 7.4 and 7.5 into 3.19 yields

� Ž .�Ž . Ž Ž . .Ý y �y s �s 1 J i , t � jJŽ i , t .� j i t j i t jŽ .7.6 y �y �i t j Ž .Ý s �s sJŽ i , t .� j i t j i t

� Ž . �ŽŽ . Ž Ž . ..Ý z �z s s �s 1 J i , t � jJŽ i , t .� j i t j i t i t j� z �z � � .i t j Ž .ž /Ý s �s sJŽ i , t .� j i t j i t


Thus, the least squares estimator of the Q�1 vector � is given by

�1� �˜ ˜ ˜Ž .�� Z Z Z yˆ ˜

where the 1�Q vector

� Ž . �ŽŽ . Ž Ž . ..Ý z �z s s �s 1 J i , t � jJŽ i , t .� j i t j i t i t jz �z �z �i t i t j Ž .Ý s �s sJŽ i , t .� j i t j i t

and the scalar

� Ž . �Ž . Ž Ž . .Ý y �y s s �s 1 J i , t � jJŽ i , t .� j i t j i t i t jy �y �y � .i t i t j Ž .Ý s �s sJŽ i , t .� j i t j i t

Finally, using either computational formula for the estimator � , we haveˆ

�1

�1�1�� Ž . Ž . Ž .7.7 � F F F y �Z � , ˆ

�J

�J

which, once again, can be computed firm-by-firm using the appropriate columns of F� and theŽ � � .appropriate rows of y �Z � , so thatˆ

Q�1

� �N s y � z �Ý Ý Ýj Ž i , t . Ž i , t . Ž i , t .q qž /ˆ Ž . Ž . q�1� J i , t � j J i , t � jj � .2 Q� s sj Ý ÝŽ i , t . Ž i , t . � �y � z � sˆŽ . Ž .J i , t � j J i , t � j Ý ÝŽ i , t . Ž i , t .q q Ž i , t .ž /Ž . q�1J i , t � j

Least Squares and Our Conditional Methods

Consider next the relation between our conditional method estimators and the conventional leastsquares estimator. Because this appendix contains the proofs of the claims in the paper, we use the

Ž .full model in equation 7.1 . The conditional method matrix Z can be expressed as

Ž .x f x s f x T s �10 f1 JŽ1 , n . 1 1 n JŽ1 , n . 1 1 1 n JŽ1 , n .11 11 11 11 11

Ž .x f x s f x T s �10 f1 JŽ1 , n . 1 1 n JŽ1 , n . 1 1 1 n JŽ1 , n .1T 1T 1T 1T 1T1 1 1 1 1

Ž .x f x s f x T s �10 fN JŽN , n . N N n JŽ1 , n . N 1 N n JŽ1 , n .N1 N 1 N 1 N 1 N 1

Ž .x f x s f x T s �10 fN JŽN , n . N N n JŽ1 , n . N 1 N n JŽ1 , n .NT NT NT NT NTN N N N N

� ��1Ž . Ž .where x are the rows of D D D XC, C is a P� Q3 matrix that selects Q3 columns of X toiŽ .place in the Z matrix, T z is the first order spline basis function defined in the text of the article,1

and all other variables are defined above. Hence, Z is N� �Q.We express the projection of F� on Z as

F ��F � �F � �Z��0 1 1 2 2


where the vector � is Q�1 and the error process � is defined as the component of the firm effectthat is orthogonal to Z�. The statistical equation substituting the projection of the firm effects forthe actual firm effects is given by

y�X��D��Z��

with an error process �� with the following properties:

� �E �� X , D , Z �0,

� � 2�var �� X , D , Z �� I ,1 N

2 Ž . �where � is the variance of � �� for all i, t that are part of N . As a direct consequence of1 i t i tthis statistical model we are assuming the following orthogonality condition:

� Ž .X F ��F � �F � �Z�0 1 1 2 2Ž .7.8 �0,� Ž .D F ��F � �F � �Z�0 1 1 2 2

which means that the columns of Z should be chosen to maximize the correlation of X and D withF ��F � �F � . We calculate the within-person least squares estimator for � and � using the0 1 1 2 2formula

�� X M X X M Z X M y� D D D� .� � �Z M X Z M Z Z M yˆ D D D�

Ž .The proof of the consistency of this estimator follows directly from the condition 7.8 so that the� � �� asymptotic distribution of � � is given by the usual least squares formulas.

Aggregation of Effects

We consider next the consequences of various aggregations and substitutions on the least squaresestimators of the various effects in the model 2.2. The algebra for all of the aggregations consideredin Section 2 is identical so we will discuss only the generic case in this appendix. An aggregation ofthe firm effect can be defined as an orthogonal decomposition of the firm effect into a part relatedto the aggregation and a part that represents the residual from this aggregation. We consider theindustry aggregation given by the matrix A and the parameters , defined in Section 2.

Ž .The model 2.2 can be restated as

�� Ž . Ž Ž . .�7.9 y�X��D��FA� I �FA A F FA A F F�� .N

If the firm effects are omitted from the model, then the statistical error becomes

�� Ž . Ž Ž . .�7.10 �� I �FA A F FA A F F�� .N

Ž Ž � � .� � � .�By construction, the design matrices FA and I �FA A F FA A F F are orthogonal. However,Nneither design matrix is orthogonal to X or D. Thus, the least squares estimates of the pure classeffects, , suffer from an excluded variable bias when they are estimated in the absence of firmeffects. Specifically, the within-person least squares estimator of the effects � � and � from

Ž . Ž .equation 7.9 with the error term defined by equation 7.10 is

�� X M X X M FA X M X��X M FA�X M �ˆ D D D D D� � .� � � � � � � � � �� A F M X A F M FA A F M X��A F M FA�A F M �D D D D D

By direct calculation of the partitioned G-inverse we have

� �� Ž Ž . .Ž Ž . .�plim ��Q A F M X X M X X M I �FA A F FA A F F�ˆ D D D NN��


where�� Ž Ž . .Q� A F M FA�A F M X X M X X M FA .D D D D

By inspection we note that the source of the inconsistency in the within-persons least squaresestimator of the class effects is the covariance between the observed characteristics, X, and the

Ž Ž � � .� � � .�part of the firm effects that is not correlated with the industry effects, I �FA A F FA A F F,Nconditional on the person effects D.

For completeness we note that if the pure class effect, say �� , is defined to be representative offirms, and not of individuals, then

�� Ž . � A A A� .

Using this definition of the pure class effect, there will be an additional term in the probability limitof �� that gives the aggregation bias associated with estimating this pure class effect using theˆfirm design matrix F and a sampling plan that is representative of persons. To our knowledge, noneof the articles cited in this paper that estimate industry or size effects from samples that arerepresentative of the population of employed individuals use a definition of a class effect that isrepresentative of the population of firms.

Firm Effects That Depend on Firm-leel Data

Suppose next that the firm effect, � , depends upon a non-time-varying characteristic of the firmover the sample period. Let the J�1 vector f contain the characteristic of firm j, less the grandmean, in each row. The grand mean should be calculated over the population of employedindividuals so that the average firm effect in the population of persons remains zero. Because theparameters of our firm effects are constant over time, we cannot nest a model of time-varying firm

Ž .characteristics in equation 2.2 . The pure firm effects can be decomposed into the part that islinearly related to f and a residual from this linear relation:

�� f�

where is a scalar parameter relating the firm’s characteristic to its firm effect and the J�1 vector gives the residual from this projection. By an argument completely analogous to the one we usedfor pure classification effects, it can be shown that the within-person least squares estimator of isalso inconsistent because X and are not orthogonal, conditional on D. Specifically,

1 �� ˆ Ž Ž . . Ž .�plim �� f FM I �X X M X X M F �� fD N D DqN��

� � � � Ž � .� �where q� f F M Ff� f F M X X M X X M Ff.D D D D

DATA APPENDIX

This Appendix contains details of the definitions of variables, missing data imputation, andstatistical calculations not reported in the text.

Education and School-Leaing Age

Our initial DAS file did not contain education information. We used supplementary informationŽavailable for a strict subsample of the DAS called the EDP, Echantillon Demographique Perma-´

.nent to impute the level of education of all other individuals in the DAS as described in Section 4.The education responses were grouped into 8 degree-level categories as shown in Data AppendixTable B1. EDP sample statistics for the men are in Data Appendix Table B2, and those for thewomen are in Data Appendix Table B3. The estimated logit equations are in Data Appendix TableB4 for men and Data Appendix Table B5 for women.


DATA APPENDIX TABLE B1

CLASSIFICATION OF FRENCH DEGREES AND U.S. EQUIVALENTS

Category Degree U.S. Equivalent

1 Sans Aucun Diplome No Terminal Degreeˆ2 CEP Elementary School

DFEO3 BEPC Junior High School

BEBEPS

Ž .4 BAC not F, G or H High SchoolBrevet superieurĆFES

Ž .5 CAP Vocational-Technical School BasicBEPEFAABAABPAFPA 1er

Ž .6 BP Vocational-Technical School AdvancedBEABECBEHBEIBESBATABAC FBAC GBAC H

7 Sante Technical College and´BTS Undergraduate UniversityDUTDESTDEULDEUSDEUG

8 2eme cycle Graduate School and Other`3eme cycle Post-Secondary Education`Grande ecoleĆAPESCAPET

ŽNotes: Authors’ adaptation of French degree codes appearing on the EDP Echantillon demographique´.permanent .



EDP SAMPLE STATISTICS�MEN

Ž .Std. Deviations in Parentheses

Degree CategoryVariable Name Overall 1 2 3 4 5 6 7 8

DOB 1925 0.188 0.254 0.295 0.160 0.136 0.055 0.098 0.063 0.186iŽ . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .0.391 0.435 0.456 0.367 0.343 0.228 0.297 0.243 0.389

1924DOB 1930 0.056 0.062 0.085 0.042 0.049 0.034 0.048 0.026 0.065iŽ . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .0.230 0.242 0.279 0.200 0.215 0.180 0.214 0.158 0.247

1929DOB 1935 0.097 0.109 0.120 0.067 0.068 0.081 0.095 0.054 0.101iŽ . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .0.296 0.311 0.325 0.250 0.252 0.273 0.293 0.226 0.301

1934DOB 1940 0.061 0.056 0.070 0.048 0.048 0.063 0.079 0.047 0.078iŽ . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .0.240 0.229 0.255 0.214 0.215 0.244 0.270 0.212 0.268

1939DOB 1945 0.094 0.070 0.091 0.075 0.098 0.117 0.133 0.118 0.149iŽ . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .0.292 0.256 0.287 0.264 0.298 0.322 0.340 0.323 0.356

1944DOB 1950 0.102 0.064 0.097 0.099 0.130 0.130 0.152 0.175 0.164iŽ . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .0.302 0.244 0.296 0.299 0.336 0.336 0.359 0.380 0.370

1949DOB 1955 0.159 0.095 0.132 0.166 0.245 0.224 0.217 0.288 0.201iŽ . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .0.365 0.293 0.339 0.372 0.430 0.417 0.412 0.453 0.401

1954DOB 1960 0.101 0.072 0.060 0.182 0.157 0.145 0.110 0.176 0.054iŽ . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .0.302 0.259 0.238 0.386 0.364 0.352 0.313 0.381 0.226

1959DOB 1977 0.141 0.218 0.050 0.160 0.069 0.151 0.068 0.052 0.003iŽ . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .0.348 0.413 0.218 0.367 0.253 0.358 0.251 0.224 0.056

Works in 0.232 0.204 0.226 0.288 0.352 0.187 0.284 0.309 0.457Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .Ile de France 0.422 0.403 0.418 0.453 0.478 0.390 0.451 0.462 0.498

CSP62 0.263 0.357 0.282 0.188 0.157 0.199 0.145 0.184 0.105Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .0.440 0.479 0.450 0.391 0.364 0.399 0.352 0.387 0.307

CSP61 0.225 0.231 0.255 0.117 0.071 0.299 0.186 0.096 0.058Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .0.418 0.422 0.436 0.321 0.266 0.458 0.390 0.295 0.233

CSP50 0.151 0.118 0.166 0.279 0.279 0.108 0.203 0.235 0.203Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .0.358 0.322 0.372 0.448 0.448 0.310 0.402 0.424 0.402

CSP40 0.112 0.061 0.110 0.173 0.233 0.080 0.258 0.275 0.225Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .0.315 0.240 0.314 0.379 0.423 0.272 0.438 0.447 0.418

CSP30 0.043 0.020 0.025 0.053 0.147 0.015 0.057 0.080 0.359Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .0.203 0.142 0.157 0.224 0.354 0.121 0.232 0.271 0.480

Number of 71229 26236 12825 3847 3036 16489 3878 2387 2531Observations



EDP SAMPLE STATISTICS�WOMEN

Ž .Std. Deviations in Parentheses

Degree CategoryVariable Name Overall 1 2 3 4 5 6 7 8

DOB 1925 0.152 0.235 0.206 0.129 0.055 0.034 0.042 0.055 0.056iŽ . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .0.359 0.424 0.405 0.336 0.229 0.181 0.202 0.228 0.230

1924DOB 1930 0.047 0.053 0.078 0.045 0.025 0.024 0.017 0.022 0.023iŽ . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .0.212 0.224 0.268 0.206 0.156 0.153 0.130 0.146 0.148

1929DOB 1935 0.084 0.096 0.118 0.070 0.043 0.061 0.054 0.049 0.052iŽ . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .0.278 0.294 0.322 0.255 0.203 0.239 0.226 0.216 0.222

1934DOB 1940 0.054 0.056 0.069 0.047 0.036 0.050 0.045 0.038 0.047iŽ . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .0.226 0.229 0.254 0.211 0.185 0.218 0.208 0.190 0.212

1939DOB 1945 0.093 0.070 0.113 0.086 0.090 0.103 0.108 0.101 0.127iŽ . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .0.290 0.255 0.317 0.281 0.287 0.304 0.311 0.301 0.334

1944DOB 1950 0.114 0.077 0.125 0.109 0.116 0.135 0.164 0.156 0.209iŽ . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .0.317 0.267 0.331 0.311 0.321 0.341 0.371 0.363 0.407

1949DOB 1955 0.186 0.112 0.180 0.167 0.285 0.247 0.252 0.298 0.354iŽ . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .0.389 0.315 0.384 0.373 0.451 0.431 0.434 0.457 0.478

1954DOB 1960 0.120 0.078 0.067 0.178 0.217 0.166 0.169 0.223 0.125iŽ . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .0.325 0.267 0.251 0.383 0.412 0.372 0.375 0.416 0.331

1959DOB 1977 0.150 0.224 0.043 0.170 0.133 0.180 0.147 0.059 0.008iŽ . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .0.357 0.417 0.202 0.375 0.339 0.384 0.355 0.236 0.088

Works in 0.254 0.237 0.239 0.286 0.333 0.221 0.316 0.283 0.466Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .Ile de France 0.435 0.425 0.426 0.452 0.471 0.415 0.465 0.451 0.499

CSP62 0.227 0.343 0.296 0.108 0.079 0.126 0.073 0.061 0.053Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .0.419 0.475 0.456 0.310 0.270 0.331 0.259 0.240 0.224

CSP61 0.050 0.061 0.067 0.027 0.023 0.044 0.027 0.029 0.015Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .0.218 0.239 0.249 0.163 0.150 0.205 0.161 0.168 0.120

CSP50 0.458 0.365 0.427 0.596 0.570 0.539 0.630 0.420 0.511Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .0.498 0.482 0.495 0.491 0.495 0.498 0.483 0.494 0.500

CSP40 0.073 0.040 0.035 0.090 0.165 0.045 0.097 0.350 0.214Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .0.261 0.195 0.185 0.286 0.371 0.208 0.296 0.477 0.410

CSP30 0.013 0.008 0.005 0.016 0.048 0.005 0.009 0.032 0.150Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .0.115 0.090 0.068 0.125 0.214 0.071 0.093 0.176 0.357

Number of 57677 19822 12768 4760 3112 10388 2633 3173 1021Observations



MULTINOMIAL LOGIT ON DEGREE CATEGORIES�MEN

Ž .Std. Errors in Parentheses

Variable Name 1 2 3 4 5 6 7

Intercept 6.254 5.828 2.465 0.803 3.985 1.714 �0.141Ž . Ž . Ž . Ž . Ž . Ž . Ž .0.122 0.125 0.134 0.142 0.125 0.139 0.158

1924DOB 1930 �0.496 �0.320 �0.333 0.005 0.392 0.266 0.102iŽ . Ž . Ž . Ž . Ž . Ž . Ž .0.105 0.106 0.131 0.133 0.113 0.132 0.179

1929DOB 1935 �0.493 �0.518 �0.344 �0.109 0.734 0.471 0.407iŽ . Ž . Ž . Ž . Ž . Ž . Ž .0.090 0.091 0.112 0.117 0.096 0.111 0.145

1934DOB 1940 �1.234 �1.117 �0.667 �0.325 0.446 0.318 0.349iŽ . Ž . Ž . Ž . Ž . Ž . Ž .0.100 0.102 0.124 0.130 0.105 0.119 0.154

1939DOB 1945 �2.031 �1.863 �1.120 �0.381 0.090 0.000 0.519iŽ . Ž . Ž . Ž . Ž . Ž . Ž .0.085 0.087 0.105 0.106 0.089 0.102 0.126

1944DOB 1950 �2.818 �2.430 �1.307 �0.379 �0.336 �0.216 0.653iŽ . Ž . Ž . Ž . Ž . Ž . Ž .0.085 0.087 0.102 0.104 0.089 0.102 0.123

1949DOB 1955 �3.388 �3.248 �1.373 �0.069 0.700 �0.363 0.843iŽ . Ž . Ž . Ž . Ž . Ž . Ž .0.086 0.089 0.100 0.101 0.090 0.103 0.121

1954DOB 1960 �2.289 �2.649 0.074 0.830 0.230 0.312 1.704iŽ . Ž . Ž . Ž . Ž . Ž . Ž .0.113 0.119 0.123 0.127 0.116 0.130 0.145

1959DOB 1977 1.897 0.246 2.891 2.855 3.319 2.742 3.339iŽ . Ž . Ž . Ž . Ž . Ž . Ž .0.360 0.363 0.364 0.369 0.362 0.368 0.379

Unskilled Blue-Collar �0.850 �1.311 �0.681 �0.193 �1.306 �0.849 �0.155Ž . Ž . Ž . Ž . Ž . Ž . Ž .at Date t in Firm 0.116 0.119 0.126 0.134 0.116 0.129 0.136

Ž .J t, tSkilled Blue-Collar �0.904 �1.074 �0.557 �0.294 �0.340 �0.006 �0.055

Ž . Ž . Ž . Ž . Ž . Ž . Ž .at Date t in Firm 0.132 0.135 0.144 0.156 0.131 0.142 0.157Ž .J i, t

Unskilled White-Collar �2.758 �2.635 �0.944 �0.217 �2.494 �1.100 �0.437Ž . Ž . Ž . Ž . Ž . Ž . Ž .at Date t in Firm 0.111 0.114 0.118 0.125 0.110 0.121 0.129

Ž .J i, tSkilled White-Collar �4.028 �3.740 �1.610 �0.377 �3.011 �1.030 �0.100


Manager at Date t �5.892 �5.996 �3.400 �1.311 �5.195 �3.036 �1.648Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .in Firm J i, t 0.124 0.132 0.142 0.136 0.131 0.141 0.148

Works in �0.627 �0.629 �0.410 �0.265 �0.766 �0.510 �0.399Ž . Ž . Ž . Ž . Ž . Ž . Ž .Ile de France 0.048 0.050 0.057 0.057 0.049 0.056 0.062



MULTINOMIAL LOGIT ON DEGREE CATEGORIES�WOMEN

Ž .Std. Errors in Parentheses

Variable Name 1 2 3 4 5 6 7

Intercept 7.296 7.148 4.645 2.263 4.555 2.693 2.278Ž . Ž . Ž . Ž . Ž . Ž . Ž .0.205 0.206 0.211 0.223 0.211 0.231 0.223

1924DOB 1930 �0.723 �0.224 �0.307 0.023 0.391 �0.148 �0.137iŽ . Ž . Ž . Ž . Ž . Ž . Ž .0.257 0.257 0.265 0.285 0.267 0.309 0.289

1929DOB 1935 �0.999 �0.683 �0.742 �0.314 0.441 0.111 �0.201iŽ . Ž . Ž . Ž . Ž . Ž . Ž .0.199 0.200 0.207 0.225 0.208 0.233 0.224

1934DOB 1940 �1.393 �1.073 �1.021 �0.383 0.371 0.054 �0.361iŽ . Ž . Ž . Ž . Ž . Ž . Ž .0.206 0.207 0.217 0.233 0.214 0.241 0.233

1939DOB 1945 �2.328 �1.743 �1.550 �0.542 �0.057 �0.210 �0.439iŽ . Ž . Ž . Ž . Ž . Ž . Ž .0.169 0.169 0.177 0.189 0.177 0.199 0.189

1944DOB 1950 �3.023 �2.429 �2.011 �0.894 �0.529 �0.461 �0.552iŽ . Ž . Ž . Ž . Ž . Ž . Ž .0.161 0.161 0.167 0.180 0.168 0.189 0.178

1949DOB 1955 �3.791 �3.433 �2.537 �0.694 �1.022 �0.927 �0.601iŽ . Ž . Ž . Ž . Ž . Ž . Ž .0.156 0.157 0.162 0.172 0.163 0.184 0.173

1954DOB 1960 �3.082 �3.323 �1.409 0.075 �0.342 �0.264 0.153iŽ . Ž . Ž . Ž . Ž . Ž . Ž .0.172 0.175 0.176 0.187 0.178 0.199 0.187

1959DOB 1977 1.070 �0.673 1.506 2.448 2.753 2.531 1.638iŽ . Ž . Ž . Ž . Ž . Ž . Ž .0.382 0.384 0.385 0.390 0.385 0.396 0.395

Unskilled Blue-Collar �0.205 �0.787 �0.778 �0.248 �0.898 �0.969 �0.511Ž . Ž . Ž . Ž . Ž . Ž . Ž .at Date t in Firm 0.195 0.196 0.202 0.210 0.196 0.212 0.213

Ž .J i, tSkilled Blue-Collar �0.634 �0.977 �0.840 �0.167 �0.645 �0.675 0.064


Unskilled White-Collar �2.250 �2.466 �1.218 �0.502 �1.593 �1.008 �0.749Ž . Ž . Ž . Ž . Ž . Ž . Ž .at Date t in Firm 0.144 0.146 0.149 0.154 0.144 0.153 0.155

Ž .J i, tSkilled White-Collar �3.853 �4.352 �2.379 �0.880 �3.272 �2.062 �0.047


Manager at Date t �5.449 �6.431 �3.977 �1.725 �5.147 �4.133 �2.052Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .in Firm J i, t 0.191 0.216 0.209 0.193 0.218 0.272 0.201

Works in �0.925 �0.983 �0.738 �0.462 �0.967 �0.541 �0.738Ž . Ž . Ž . Ž . Ž . Ž . Ž .Ile de France 0.069 0.070 0.074 0.076 0.070 0.078 0.077


Seniority and Labor Force Experience

In order to impute a level of seniority for left-censored employment spells, we ran regressionsŽ .separately for men and women of seniority on a set of demographic and occupational characteris-

Ž .tics using data from the 1978 Salary Structure Survey ESS, Enquete sur la Structure des Salaires .ˆŽ . Ž .The results for men are shown in equation 8.1 and the results for women are in 8.2 . All

regressions included controls for 84 industries.

Ž .8.1 seniority � 2.513i tŽ .0.081

� � � �� 14.151 DOB �1924 � 12.820 1925�DOB �1929i iŽ . Ž .0.067 0.067

� � � �� 10.299 1930�DOB �1934 � 7.445 1935�DOB �1939i iŽ . Ž .0.066 0.067

� � � �� 4.748 1940�DOB �1944 � 2.569 1945�DOBs�1949iŽ . Ž .0.067 0.065

� � � �� 0.612 1950�DOB �1954 � 0.642 1955�DOB �1959i iŽ . Ž .0.065 0.067

� 4.039 CSP30 � 4.939 CSP40i t i tŽ . Ž .0.038 0.031

� 1.885 CSP50 � 2.898 CSP61i t i tŽ . Ž .0.037 0.027

� 0.958 Ile de France ,i tŽ .0.026

N�547,746, R2 �0.461,

Ž .8.2 seniority � 2.114i tŽ .0.084

� � � �� 12.669 DOB �1924 � 11.014 1925�DOB �1929i iŽ . Ž .0.074 0.075

� � � �� 8.979 1930�DOB �1934 � 7.278 1935�DOB �1939i iŽ . Ž .0.073 0.074

� � � �� 5.989 1940�DOB �1944 � 4.604 1945�DOB �1949i iŽ . Ž .0.075 0.070

� � � �� 2.822 1950�DOB �1954 � 0.641 1955�DOB �1959i iŽ . Ž .0.068 0.068

� 5.116 CSP30 � 5.789 CSP40i t i tŽ . Ž .0.082 0.057

� 1.442 CSP50 � 2.429 CSP61i t i tŽ . Ž .0.037 0.054

� 0.988 Ile de France ,i tŽ .0.031

N�260,580, R2 �0.373,


where

DOB �Date of birth of individual i ,i

CSP30 �1 if i is an engineer, professional, or manager,i t

CSP40 �1 if i is technician or technical white-collar,i tŽ .8.3

CSP50 �1 if i is any other white-collar,i t

CSP61 �1 if i is a skilled blue-collar,i t

Ž .CSP62 �1 if i is an unskilled blue-collar omitted ,i t

Ile de France �1 if the establishment is in Ile-de-France.i t

The excluded date of birth category was 1960�DOB . The coefficients on the industry indicators areinot shown.

To compute the values of seniority and labor force experience, we used the following algorithms.If the individual was left-censored and the imputed job seniority was negative, we set job seniorityprior to 1976 to zero. If the individual was first observed after 1976, we assumed that job seniority onthat job prior to the date of the first DAS observation for the individual was zero. If the age at the

Ž .date of any observation 1976 or otherwise was less than the expected school-leaving age, both totalŽlabor force experience and prior job seniority were set to zero. In all other cases when the age was

.greater than the expected school-leaving age , we calculated total labor market experience and jobseniority as follows. If the observation was the earliest appearance of the individual in our data, weset job seniority equal to job seniority up to the date of the first observation plus the number of daysworked for that enterprise in the year of the first observation, divided by 360 and we set total labormarket experience to the current age less the school-leaving age. If the observation was not the firstfor the individual but there was an observation in the previous year for the person,44 we added 1 tototal labor market experience. If the individual was employed for the majority of the current year bythe same enterprise that employed him or her for the majority of the previous year, i.e. SIREN �tSIREN , we added 1 to the level of seniority at t�1. If SIREN �SIREN , we set seniority equalt�1 t t�1to the number of days worked divided by 360.

If, on the other hand, there was no observation in the previous year, we distinguished betweent�1982 or t�1984 and other years. When t�1982 or 1984, total labor market experience was

Ž .increased by 1 reflecting experience gained in the year of the observation . If the current SIRENand the most recent previous SIREN were the same, we added the number of days worked dividedby 360 to the most recent previous level of seniority. This is similar to assuming that the worker wastemporarily laid off but retained his or her seniority in the firm when recalled. Otherwise, we setseniority to the number of days worked divided by 360.

ŽIn the case where t�1982 or t�1984, if the preceding observation was 2 years earlier i.e. the.missing data only occurred over a period when no data were available for any individual , we

increased total labor market experience by 2. If SIREN �SIREN , seniority was increased by 2. Ift�2 tSIREN �SIREN , seniority was increased by 0.5 plus the number of days worked divided by 360.45

t�2 t

44 ŽThe structure of our database is such that this condition observations for individual i at both t.and t�1 could only fail to be satisfied under 3 conditions. The first is that the individual was

employed in the civil service in the intervening years. The second is that the individual wasunemployed for an entire calendar year. The third is that t�1982 or t�1984, since we were notgiven access to the data for 1981 or 1983. We largely discount the first possibility for the reasonsmentioned in the text. The other two possibilities are treated explicitly.

45 We assumed that the probability the individual was reemployed in the missing year was equalto the probability that the individual was reemployed in the observation year. Thus, the expected

1 1Ž . Ž .increment to job seniority is the share of the year worked in the observation year plus 0 � 12 2

�0.5.


If the preceding observation was more than 2 years earlier, we increased total labor marketexperience by 1.5.46 If the current SIREN and the most recent previous SIREN were the same, weadded the number of days worked divided by 360 plus 0.5 to the most recent previous level ofseniority. This is similar to assuming that the worker was recalled from temporary layoff with equalprobability in the observation year and in the missing year. If the two SIRENs were different, we setseniority to 0.5 plus the number of days worked divided by 360.

Elimination of Outliers

ŽWe ran a standard log earnings regression the dependent variable was the logarithm of realannualized compensation cost, LFRAISRE, the same one used in the analyses reported in Tables

.II�XII on our DAS data and considered all observations that were more than 5 standard deviationsaway from their predicted values as outliers. These observations were discarded. The estimated

Ž .coefficients of this earnings regression are shown in equation 8.4 .

Ž .8.4 LFRAISRE �� 3.250i tŽ .0.005

� 0.210 Male � 0.123 Ile de Francei i tŽ . Ž .0.000 0.000

� 0.082 Year � 0.056 Degree Category 2it iŽ . Ž .0.000 0.002

� 0.415 Degree Category 3 � 0.627 Degree Category 4i iŽ . Ž .0.002 0.003



� 0.055 Experience � 0.222 Experience2i t i t

Ž . Ž .0.000 0.003

� 0.052 Experience3 � 0.005 Experience4 ,i t i tŽ . Ž .0.001 0.000

N�5,325,352, R2 �0.437, ��0.477.

Definition of Z Variables and Coefficients in the Conditional Method

Data Appendix Table B6 contains the definitions, regression coefficients, and coefficient standardŽ .errors for the Z variables used in estimating the statistical model 3.17 as reported in Table III in

the column labelled ‘‘Conditional Method Persons First.’’

Pooled Regression for Order-Dependent Persons-First Estimation

Recovery of the firm effects was done in the conditional methods on a firm-by-firm basis. Allobservations corresponding to firms for which there were fewer than 10 observations were groupedtogether and included in a single, pooled regression. The results of this regression for the pooled

Ž .‘‘firm’’ in the order-dependent, persons-first case are shown in equation 8.5 . The results for theorder-independent pooled ‘‘firm’’ are not shown.

Ž . Ž .8.5 DLFRAISR �� 0.028 � 0.003 s � 0.005 T s �10 ,i t i t 1 i tŽ . Ž . Ž .3.375e-4 8.476e-5 1.772 e-4

N�1,353,794, R2 �0.0013.

46 We assumed that the probability the individual was reemployed in the missing year was equalto the probability that the individual was reemployed in the observation year. Thus, the expected

1 1Ž . Ž .increment to total labor market experience is 1 � 2 �1.5.2 2



SUMMARY STATISTICS, COEFFICIENTS AND STANDARD ERRORS FOR Z VARIABLES

IN THE CONDITIONAL METHOD ORDER INDEPENDENT ESTIMATION

Standard StandardVariable Definition Mean Deviation Coefficient Error

Firm size� average experience 2.54E-05 3.16E-06 1.11E-05 3.82E-06Firm size � age at end of school 1.79E-04 2.91E-06 1.77E-05 3.58E-06Firm size squared � average experience �7.57E-08 2.00E-08 6.38E-08 2.00E-08Firm size squared � age at end of school �5.28E-07 1.00E-08 �3.06E-08 2.00E-08Firm size � seniority � average experience 3.23E-06 3.30E-07 2.76E-06 3.50E-07Firm size � seniority � age at end of school �1.43E-05 4.50E-07 �6.95E-06 4.30E-07Firm size squared � seniority � average �5.76E-09 1.60E-07 �1.12E-09 7.19E-10

experienceFirm size squared � seniority � age at 4.47E-08 1.00E-08 2.01E-08 1.91E-07

end of schoolIndustry 1 � average experience �3.92E-04 1.28E-04 �2.06E-03 3.19E-03Industry 1� age at end of school �2.22E-02 1.48E-04 1.04E-02 3.49E-03Industry 1 � seniority � average experience 3.95E-04 1.61E-05 1.50E-04 1.53E-05Industry 1 � seniority � age at end of school �2.98E-04 2.51E-05 �1.12E-04 2.17E-05Industry 2 � average experience 2.10E-03 2.17E-04 �4.71E-03 3.20E-03Industry 2 � age at end of school 1.62E-02 2.20E-04 1.39E-02 3.50E-03Industry 2 � seniority � average experience �1.25E-04 2.18E-05 �1.41E-04 2.27E-05Industry 2 � seniority � age at end of school 6.14E-04 3.14E-05 3.32E-04 3.04E-05Industry 3 � average experience 3.82E-04 7.75E-05 �1.93E-03 3.19E-03Industry 3 � age at end of school �3.61E-02 8.33E-05 1.03E-02 3.49E-03Industry 3 � seniority � average experience 2.07E-04 8.98E-06 8.41E-05 8.00E-06Industry 3 � seniority � age at end of school �4.80E-05 1.36E-05 �1.52E-05 1.14E-05Industry 4 � average experience �2.52E-04 7.46E-05 �2.15E-03 3.19E-03Industry 4 � age at end of school �1.76E-02 7.08E-05 1.08E-02 3.49E-03Industry 4� seniority � average experience 4.09E-05 8.03E-06 8.92E-05 7.62E-06Industry 4 � seniority � age at end of school 3.66E-04 1.12E-05 �1.38E-05 9.93E-06Industry 5 � average experience 2.16E-03 8.12E-05 �1.95E-03 3.19E-03Industry 5 � age at end of school �3.59E-02 8.61E-05 9.18E-03 3.49E-03Industry 5 � seniority � average experience 2.92E-04 9.78E-06 1.14E-04 9.37E-06Industry 5 � seniority � age at end of school �4.70E-04 1.48E-05 7.38E-06 1.29E-05Industry 6 � average experience 1.02E-03 8.46E-05 1.67E-04 3.19E-03Industry 6 � age at end of school �2.94E-02 1.05E-04 4.62E-03 3.49E-03Industry 6 � seniority � average experience 7.20E-04 1.20E-05 1.07E-04 1.07E-05Industry 6 � seniority � age at end of school �1.41E-03 1.85E-05 �1.00E-04 1.50E-05Industry 7 � average experience �3.46E-04 6.93E-05 �2.16E-03 3.19E-03Industry 7 � age at end of school 6.53E-03 8.00E-05 8.91E-03 3.49E-03Industry 7 � seniority � average experience �4.73E-05 9.20E-06 �4.40E-05 8.55E-06Industry 7� seniority � age at end of school 9.89E-04 1.38E-05 2.34E-04 1.17E-05Industry 8 � average experience �3.60E-04 1.32E-04 �2.88E-03 3.19E-03Industry 8 � age at end of school 2.35E-02 1.39E-04 9.77E-03 3.49E-03Industry 8 � seniority � average experience �3.22E-04 1.53E-05 7.68E-05 1.49E-05Industry 8 � seniority � age at end of school 1.70E-03 2.25E-05 1.02E-04 2.06E-05Industry 9 � average experience 5.22E-04 5.53E-05 �2.81E-03 3.19E-03Industry 9 � age at end of school 3.57E-02 5.89E-05 8.25E-03 3.49E-03Industry 9 � seniority � average experience �3.87E-04 6.81E-06 �2.85E-05 6.40E-06Industry 9 � seniority � age at end of school 1.89E-03 9.63E-06 1.79E-04 8.36E-06Industry 10 � average experience �1.98E-03 8.29E-05 �3.20E-03 3.19E-03Industry 10 � age at end of school 3.43E-02 7.92E-05 8.87E-03 3.49E-03Industry 10 � seniority � average experience �1.10E-04 9.56E-06 �1.97E-05 1.01E-05Industry 10 � seniority � age at end of school 0.001673 0.00001264 0.000238 0.00001243

Notes: These coefficients supplement the coefficients reported in Table III, Column ‘‘Conditional Method Persons First.’’



DESCRIPTIVE STATISTICS FOR BASIC INDIVIDUAL LEVEL VARIABLES BY SEX FOR 1976 TO 1987

Men WomenStandard Standard

Variable Definition Mean Deviation Mean Deviation

Real Total Annual Compensation Cost, 89.0967 61.6302 67.3646 37.42081,000FF 1980ŽLog Real Annual Compensation Cost, 4.3442 0.5187 4.0984 0.4801

.1980 FFTotal Labor Force Experience 17.2531 11.8258 15.4301 12.0089

2Ž .Total Labor Force Experience 100 4.3752 4.9197 3.8230 4.94403Ž .Total Labor Force Experience 1,000 13.1530 19.4305 11.6079 19.68634Ž .Total Labor Force Experience 10,000 43.3453 77.9542 39.0589 80.3251

Seniority 7.7067 7.5510 6.5437 6.5268Lives in Ile-de-France 0.2561 0.2910Ž .Paris Metropolitan Region

No Known Degree 0.3064 0.2190 0.2971 0.2124Completed Elementary School 0.1556 0.1458 0.1893 0.1739Completed Junior High School 0.0565 0.0792 0.0869 0.1008

Ž .Completed High School Baccalaureat 0.0528 0.0804 0.0711 0.0881´Basic Vocational-Technical Degree 0.2652 0.1849 0.1926 0.1545Advanced Vocational-Technical Degree 0.0701 0.0893 0.0532 0.0802Technical College or University Diploma 0.0469 0.0754 0.0838 0.1247Graduate School Diploma 0.0465 0.0964 0.0259 0.0551Year of data 81.3106 3.7250 81.4730 3.7180Number of Observations for the 4,402.3800 16,164.6200 1,605.3100 7,797.1300

Firm in SampleObservations 3,434,530 1,870,578Persons 711,518 454,787Proportion with Identified

Least Squares Estimate ofIndividual and Firm Effect 0.7425 0.7448

Ž .Source: Authors’ calculations based on the Declarations annuelles des salaires DAS .´

Construction of the Operating Income Variable

Ž .The operating income variable excedent brut d’exploitation was constructed as in the followingequation:

Ž . Ž .8.6 EBE�ventes de marchandises merchandise soldŽ .�achat de marchandises merchandise purchased

�variation de stock de marchandisesŽ .variation in merchandise inventory

Ž .�ventes de biens goods sold

Ž .�ventes de services services soldŽ .�production stockee inventoried production´

Ž .�production immobilisee unfinished production´Ž .�achats de matieres premieres primary materials purchased` `

�variation de stocks sur matieres premieres` `Ž .variation of primary materials inventories


�autres achats et charges externes

Ž .other purchases and outside charges

Ž .�subventions d’exploitation incentives for production

�impots, taxes et versements assimilesˆ ´

Žvalue added tax and other accrued taxes on

.or credits for production

Ž .�salaires et traitements salaries and benefits

Ž .�charges sociales payroll taxes .

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Documents

Econometrica, Vol. 67, No. 2 March, 1999 , 251 Ž. 333